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Zero and negative exponents

Practice evaluating numbers and expressions with zero and negative exponents.

Any nonzero number raised to the zero power equals $1$:

If $a \neq 0$, then $a^0 = 1$.

To raise a nonzero number to a negative power, take the reciprocal of the number raised to the corresponding positive power:

$a^{-n} = \frac{1}{a^n}$.

If the base is a fraction and the exponent is negative, take the reciprocal of the fraction and make the exponent positive:

$\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n, \quad a \neq 0, \quad b \neq 0$

$\left(\frac{2}{5}\right)^{-3} = \left(\frac{5}{2}\right)^3 = \frac{125}{8}$

A negative exponent does not necessarily produce a negative result. The sign depends on the base and the exponent.

$(-2)^{-2} = \frac{1}{(-2)^2} = \frac{1}{4}$

$(-2)^{-3} = \frac{1}{(-2)^3} = -\frac{1}{8}$

If a negative number is raised to an even power, the result is positive. If a negative number is raised to an odd power, the result is negative. This rule also holds for negative exponents.

First identify the base, apply the zero- or negative-exponent rule, and then simplify the numerical or algebraic expression.

Review the lesson on zero and negative exponents.

{"questions":[{"content":"$$5^0$$[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$1$","$5$","$0$","$-5$"],"placeholder":0,"answer":0}},"step":1,"hints":["According to the rule $a^0=1$, if $a\\neq0$.","Accordingly, $5\\neq0$ and $5^0=1$."]}]}

{"questions":[{"content":"$$(-3)^0$$[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$1$","$0$","$-3$","$3$"],"placeholder":0,"answer":0}},"step":1,"hints":["Let's remember the rule: $a^0=1$, if $a\\neq 0$.","The base is equal to $-3$, this number is not zero, which means $(-3)^0=1$."]},{"content":"$$5^0$$[[fill_choice_big-2]]","widgets":{"fill_choice_big-2":{"type":"fill_choice_big","options":["$0$","$1$","$5$","$-1$"],"placeholder":0,"answer":1}},"step":1,"hints":["By the definition of a power: $a^0=1$ when $a\\neq 0$.","Since $5\\neq 0$, we get $5^0=1$."]},{"content":"$$(-7)^0$$[[fill_choice_big-3]]","widgets":{"fill_choice_big-3":{"type":"fill_choice_big","options":["$-7$","$7$","$0$","$1$"],"placeholder":0,"answer":3}},"step":1,"hints":["We use the rule: $a^0=1$ with $a\\neq 0$.","The base is $-7\\neq 0$, which means $(-7)^0=1$."]},{"content":"$$12^0$$[[fill_choice_big-4]]","widgets":{"fill_choice_big-4":{"type":"fill_choice_big","options":["$12$","$1$","$0$","$-12$"],"placeholder":0,"answer":1}},"step":1,"hints":["To the power of zero, any non-zero number gives 1: $a^0=1$.","Since $12\\neq 0$, then $12^0=1$."]},{"content":"$$\\left(-\\frac{2}{3}\\right)^0$$[[fill_choice_big-5]]","widgets":{"fill_choice_big-5":{"type":"fill_choice_big","options":["$\\frac{2}{3}$","$-\\frac{2}{3}$","$1$","$0$"],"placeholder":0,"answer":2}},"step":1,"hints":["Zero exponent: $a^0=1$, if $a\\neq 0$.","The base is $-\\frac{2}{3}\\neq 0$, which means the result is $1$."]},{"content":"$$\\left(\\frac{9}{10}\\right)^0$$[[fill_choice_big-6]]","widgets":{"fill_choice_big-6":{"type":"fill_choice_big","options":["$0$","$1$","$\\frac{9}{10}$","$\\frac{10}{9}$"],"placeholder":0,"answer":1}},"step":1,"hints":["For a zero exponent: $a^0=1$ when $a\\neq 0$.","The base of $\\frac{9}{10}$ is not equal to zero, which means the value is equal to $1$."]},{"content":"$$101^0$$[[fill_choice_big-7]]","widgets":{"fill_choice_big-7":{"type":"fill_choice_big","options":["$101$","$0$","$1$","$-101$"],"placeholder":0,"answer":2}},"step":1,"hints":["Any non-zero number to the zero power is equal to $1$: $a^0=1$.","Since $101\\neq 0$, then $101^0=1$."]},{"content":"$$(-1)^0$$[[fill_choice_big-8]]","widgets":{"fill_choice_big-8":{"type":"fill_choice_big","options":["$-1$","$1$","$0$","$-0$"],"placeholder":0,"answer":1}},"step":1,"hints":["Apply the rule: $a^0=1$ if $a\\neq 0$.","The base is $-1\\neq 0$, which means $(-1)^0=1$."]},{"content":"$$2^0$$[[fill_choice_big-9]]","widgets":{"fill_choice_big-9":{"type":"fill_choice_big","options":["$0$","$2$","$1$","$-2$"],"placeholder":0,"answer":2}},"step":1,"hints":["Let's remember: $a^0=1$ with $a\\neq 0$.","Since $2\\neq 0$, then $2^0=1$."]},{"content":"$$(-25)^0$$[[fill_choice_big-10]]","widgets":{"fill_choice_big-10":{"type":"fill_choice_big","options":["$-25$","$25$","$1$","$0$"],"placeholder":0,"answer":2}},"step":1,"hints":["For any non-zero base the following is true: $a^0=1$.","The base is $-25\\neq 0$, which means $(-25)^0=1$."]}],"mix":1}

{"questions":[{"content":"$$5^{-3}$$[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$\\frac{1}{125}$","$-\\frac{1}{125}$","$125$","$-125$"],"placeholder":0,"answer":0}},"step":1,"hints":["According to the rule $a^{-n}=\\frac{1}{a^n}$ at $a\\neq0$.","Accordingly:<br />$5^{-3}=\\frac{1}{5^3}=\\frac{1}{125}$."]}]}

{"questions":[{"content":"$$2^{-5}$$[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$\\frac{1}{32}$","$-\\frac{1}{32}$","$32$","$-32$"],"placeholder":0,"answer":0}},"step":1,"hints":["According to the rule $a^{-n}=\\frac{1}{a^n}$ at $a\\neq0$.","Then $2^{-5}=\\frac{1}{2^5}=\\frac{1}{32}$."]},{"content":"$$3^{-4}$$[[fill_choice_big-2]]","widgets":{"fill_choice_big-2":{"type":"fill_choice_big","options":["$\\frac{1}{81}$","$-\\frac{1}{81}$","$81$","$-81$"],"placeholder":0,"answer":0}},"step":1,"hints":["Use the $a^{-n}=\\frac{1}{a^n}$ rule when $a\\neq0$.","We get $3^{-4}=\\frac{1}{3^4}=\\frac{1}{81}$."]},{"content":"$$4^{-3}$$[[fill_choice_big-3]]","widgets":{"fill_choice_big-3":{"type":"fill_choice_big","options":["$\\frac{1}{64}$","$-\\frac{1}{64}$","$64$","$-64$"],"placeholder":0,"answer":0}},"step":1,"hints":["Apply the negative exponent rule: $a^{-n}=\\frac{1}{a^n}$.","We have $4^{-3}=\\frac{1}{4^3}=\\frac{1}{64}$."]},{"content":"$$6^{-2}$$[[fill_choice_big-4]]","widgets":{"fill_choice_big-4":{"type":"fill_choice_big","options":["$\\frac{1}{36}$","$-\\frac{1}{36}$","$36$","$-36$"],"placeholder":0,"answer":0}},"step":1,"hints":["Apply the formula $a^{-n}=\\frac{1}{a^n}$.","Then $6^{-2}=\\frac{1}{6^2}=\\frac{1}{36}$."]},{"content":"$$10^{-3}$$[[fill_choice_big-5]]","widgets":{"fill_choice_big-5":{"type":"fill_choice_big","options":["$\\frac{1}{1000}$","$-\\frac{1}{1000}$","$1000$","$-1000$"],"placeholder":0,"answer":0}},"step":1,"hints":["Remember: $a^{-n}=\\frac{1}{a^n}$.","We get $10^{-3}=\\frac{1}{10^3}=\\frac{1}{1000}$."]},{"content":"$$12^{-2}$$[[fill_choice_big-6]]","widgets":{"fill_choice_big-6":{"type":"fill_choice_big","options":["$\\frac{1}{144}$","$-\\frac{1}{144}$","$144$","$-144$"],"placeholder":0,"answer":0}},"step":1,"hints":["According to the rule: $a^{-n}=\\frac{1}{a^n}$.","Then $12^{-2}=\\frac{1}{12^2}=\\frac{1}{144}$."]},{"content":"$$(-2)^{-4}$$[[fill_choice_big-7]]","widgets":{"fill_choice_big-7":{"type":"fill_choice_big","options":["$\\frac{1}{16}$","$-\\frac{1}{16}$","$16$","$-16$"],"placeholder":0,"answer":0}},"step":1,"hints":["Cast $a^{-n}=\\frac{1}{a^n}$ first.","Next $(-2)^{-4}=\\frac{1}{(-2)^4}=\\frac{1}{16}$, since an even power gives a positive number."]},{"content":"$$(-3)^{-3}$$[[fill_choice_big-8]]","widgets":{"fill_choice_big-8":{"type":"fill_choice_big","options":["$-\\frac{1}{27}$","$\\frac{1}{27}$","$-27$","$27$"],"placeholder":0,"answer":0}},"step":1,"hints":["Use $a^{-n}=\\frac{1}{a^n}$.","We get $(-3)^{-3}=\\frac{1}{(-3)^3}=\\frac{1}{-27}=-\\frac{1}{27}$ because the exponent is odd."]},{"content":"$$(-5)^{-2}$$[[fill_choice_big-9]]","widgets":{"fill_choice_big-9":{"type":"fill_choice_big","options":["$\\frac{1}{25}$","$-\\frac{1}{25}$","$25$","$-25$"],"placeholder":0,"answer":0}},"step":1,"hints":["Apply the formula $a^{-n}=\\frac{1}{a^n}$.","Then $(-5)^{-2}=\\frac{1}{(-5)^2}=\\frac{1}{25}$."]},{"content":"$$8^{-1}$$[[fill_choice_big-10]]","widgets":{"fill_choice_big-10":{"type":"fill_choice_big","options":["$\\frac{1}{8}$","$-\\frac{1}{8}$","$8$","$-8$"],"placeholder":0,"answer":0}},"step":1,"hints":["According to the rule: $a^{-1}=\\frac{1}{a}$ with $a\\neq0$.","So $8^{-1}=\\frac{1}{8}$."]},{"content":"$$7^{-2}$$[[fill_choice_big-11]]","widgets":{"fill_choice_big-11":{"type":"fill_choice_big","options":["$\\frac{1}{49}$","$-\\frac{1}{49}$","$49$","$-49$"],"placeholder":0,"answer":0}},"step":1,"hints":["Use the $a^{-n}=\\frac{1}{a^n}$ rule.","We get $7^{-2}=\\frac{1}{7^2}=\\frac{1}{49}$."]},{"content":"$$9^{-2}$$[[fill_choice_big-12]]","widgets":{"fill_choice_big-12":{"type":"fill_choice_big","options":["$\\frac{1}{81}$","$-\\frac{1}{81}$","$81$","$-81$"],"placeholder":0,"answer":0}},"step":1,"hints":["Apply the formula $a^{-n}=\\frac{1}{a^n}$.","Then $9^{-2}=\\frac{1}{9^2}=\\frac{1}{81}$."]},{"content":"$$25^{-2}$$[[fill_choice_big-13]]","widgets":{"fill_choice_big-13":{"type":"fill_choice_big","options":["$\\frac{1}{625}$","$-\\frac{1}{625}$","$625$","$-625$"],"placeholder":0,"answer":0}},"step":1,"hints":["Use the negative power rule: $a^{-n}=\\frac{1}{a^n}$.","We have $25^{-2}=\\frac{1}{25^2}=\\frac{1}{625}$."]},{"content":"$$0.2^{-2}$$[[fill_choice_big-14]]","widgets":{"fill_choice_big-14":{"type":"fill_choice_big","options":["$25$","$-25$","$\\frac{1}{25}$","$-\\frac{1}{25}$"],"placeholder":0,"answer":0}},"step":1,"hints":["According to the rule: $a^{-n}=\\frac{1}{a^n}$.","Then $0.2^{-2}=\\frac{1}{(0.2)^2}$. Since $0.2=\\frac{1}{5}$, we get $\\frac{1}{(\\frac{1}{5})^2}=\\frac{1}{\\frac{1}{25}}=25$."]},{"content":"$$\\left(\\frac{2}{3}\\right)^{-3}$$[[fill_choice_big-15]]","widgets":{"fill_choice_big-15":{"type":"fill_choice_big","options":["$\\frac{27}{8}$","$\\frac{8}{27}$","$-\\frac{27}{8}$","$-\\frac{8}{27}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Use the rule: $a^{-n}=\\frac{1}{a^n}$ or flip the fraction: $\\left(\\frac{m}{n}\\right)^{-k}=\\left(\\frac{n}{m}\\right)^k$.","We have $\\left(\\frac{2}{3}\\right)^{-3}=\\left(\\frac{3}{2}\\right)^3=\\frac{27}{8}$."]}],"mix":1}

{"questions":[{"content":"$$\\left(\\frac{3}{5}\\right)^{-2}$$[[fill_choice_big-16]]","widgets":{"fill_choice_big-16":{"type":"fill_choice_big","options":["$\\frac{25}{9}$","$\\frac{9}{25}$","$-\\frac{25}{9}$","$-\\frac{9}{25}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Use the rule: $\\left(\\frac{m}{n}\\right)^{-k}=\\left(\\frac{n}{m}\\right)^k$.","Then $\\left(\\frac{3}{5}\\right)^{-2}=\\left(\\frac{5}{3}\\right)^2=\\frac{25}{9}$."]},{"content":"$$\\left(\\frac{4}{7}\\right)^{-3}$$[[fill_choice_big-17]]","widgets":{"fill_choice_big-17":{"type":"fill_choice_big","options":["$\\frac{343}{64}$","$\\frac{64}{343}$","$-\\frac{343}{64}$","$-\\frac{64}{343}$"],"placeholder":0,"answer":0}},"step":1,"hints":["A negative exponent means taking the reciprocal of the fraction: $\\left(\\frac{m}{n}\\right)^{-k}=\\left(\\frac{n}{m}\\right)^k$.","We get $\\left(\\frac{4}{7}\\right)^{-3}=\\left(\\frac{7}{4}\\right)^3=\\frac{343}{64}$."]},{"content":"$$\\left(\\frac{5}{2}\\right)^{-4}$$[[fill_choice_big-18]]","widgets":{"fill_choice_big-18":{"type":"fill_choice_big","options":["$\\frac{16}{625}$","$\\frac{625}{16}$","$-\\frac{16}{625}$","$-\\frac{625}{16}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Apply the rule: $\\left(\\frac{m}{n}\\right)^{-k}=\\left(\\frac{n}{m}\\right)^k$.","Then $\\left(\\frac{5}{2}\\right)^{-4}=\\left(\\frac{2}{5}\\right)^4=\\frac{16}{625}$."]},{"content":"$$\\left(\\frac{7}{9}\\right)^{-2}$$[[fill_choice_big-19]]","widgets":{"fill_choice_big-19":{"type":"fill_choice_big","options":["$\\frac{81}{49}$","$\\frac{49}{81}$","$-\\frac{81}{49}$","$-\\frac{49}{81}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Negative exponent of a fraction: invert the fraction and raise it to a positive exponent.","We have $\\left(\\frac{7}{9}\\right)^{-2}=\\left(\\frac{9}{7}\\right)^2=\\frac{81}{49}$."]},{"content":"$$\\left(\\frac{11}{3}\\right)^{-2}$$[[fill_choice_big-20]]","widgets":{"fill_choice_big-20":{"type":"fill_choice_big","options":["$\\frac{9}{121}$","$\\frac{121}{9}$","$-\\frac{9}{121}$","$-\\frac{121}{9}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Use the rule: $\\left(\\frac{m}{n}\\right)^{-k}=\\left(\\frac{n}{m}\\right)^k$.","Then $\\left(\\frac{11}{3}\\right)^{-2}=\\left(\\frac{3}{11}\\right)^2=\\frac{9}{121}$."]},{"content":"$$\\left(\\frac{8}{5}\\right)^{-3}$$[[fill_choice_big-21]]","widgets":{"fill_choice_big-21":{"type":"fill_choice_big","options":["$\\frac{125}{512}$","$\\frac{512}{125}$","$-\\frac{125}{512}$","$-\\frac{512}{125}$"],"placeholder":0,"answer":0}},"step":1,"hints":["First, reverse the fraction: $\\left(\\frac{8}{5}\\right)^{-3}=\\left(\\frac{5}{8}\\right)^3$.","Next $\\left(\\frac{5}{8}\\right)^3=\\frac{125}{512}$."]},{"content":"$$\\left(\\frac{13}{4}\\right)^{-2}$$[[fill_choice_big-22]]","widgets":{"fill_choice_big-22":{"type":"fill_choice_big","options":["$\\frac{16}{169}$","$\\frac{169}{16}$","$-\\frac{16}{169}$","$-\\frac{169}{16}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Use: $\\left(\\frac{m}{n}\\right)^{-k}=\\left(\\frac{n}{m}\\right)^k$.","Then $\\left(\\frac{13}{4}\\right)^{-2}=\\left(\\frac{4}{13}\\right)^2=\\frac{16}{169}$."]},{"content":"$$\\left(\\frac{2}{9}\\right)^{-3}$$[[fill_choice_big-23]]","widgets":{"fill_choice_big-23":{"type":"fill_choice_big","options":["$\\frac{729}{8}$","$\\frac{8}{729}$","$-\\frac{729}{8}$","$-\\frac{8}{729}$"],"placeholder":0,"answer":0}},"step":1,"hints":["For a negative exponent, take the reciprocal of the fraction: $\\left(\\frac{2}{9}\\right)^{-3}=\\left(\\frac{9}{2}\\right)^3$.","Then $\\left(\\frac{9}{2}\\right)^3=\\frac{9^3}{2^3}=\\frac{729}{8}$."]},{"content":"$$\\left(\\frac{6}{11}\\right)^{-2}$$[[fill_choice_big-24]]","widgets":{"fill_choice_big-24":{"type":"fill_choice_big","options":["$\\frac{121}{36}$","$\\frac{36}{121}$","$-\\frac{121}{36}$","$-\\frac{36}{121}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Use the rule: $\\left(\\frac{m}{n}\\right)^{-k}=\\left(\\frac{n}{m}\\right)^k$.","Then $\\left(\\frac{6}{11}\\right)^{-2}=\\left(\\frac{11}{6}\\right)^2=\\frac{121}{36}$."]},{"content":"$$\\left(\\frac{15}{8}\\right)^{-2}$$[[fill_choice_big-25]]","widgets":{"fill_choice_big-25":{"type":"fill_choice_big","options":["$\\frac{64}{225}$","$\\frac{225}{64}$","$-\\frac{64}{225}$","$-\\frac{225}{64}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Take the reciprocal of the fraction and make the exponent positive: $\\left(\\frac{15}{8}\\right)^{-2}=\\left(\\frac{8}{15}\\right)^2$.","We get $\\left(\\frac{8}{15}\\right)^2=\\frac{64}{225}$."]}],"mix":1}

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