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Exponents
Exponents with fractional bases

If the base of an exponential expression is a fraction, the exponent tells us how many times to write that fraction as a factor.

 $(\frac{2}{3})^2=\frac{2}{3}\cdot\frac{2}{3}=\frac{2\cdot2}{3\cdot3}=\frac{4}{9}$ Since the exponent is 2, write the base, $\frac{2}{3}$, as a factor 2 times.

Evaluate $(\frac{1}{4})^3$

 $(\frac{1}{4})^3=\frac{1}{4}\cdot\frac{1}{4}\cdot\frac{1}{4}$ Since the exponent is 3, write the base, $\frac{1}{4}$, as a factor 3 times. $=\frac{1\cdot1\cdot1}{4\cdot4\cdot4}$ Multiply the numerators Multiply the denominators $=\frac{1}{64}$

We read $(\frac{1}{4})^3$ as "one-fourth raised to the third power", or as "one fourth, cubed"

Evaluate $(-\frac{2}{3})^2$

 $(-\frac{2}{3})^2=(-\frac{2}{3})\cdot(-\frac{2}{3})$ Since the exponent is 2, write the base, $-\frac{2}{3}$, as a factor 2 times. $=\frac{2\cdot2}{3\cdot3}$ Multiply the numerators Multiply the denominators $=\frac{4}{9}$

We read $(-\frac{2}{3})^2$ as "negative two-thirds raised to the second power", or as "negative two-thirds, squared"

Evaluate $-(\frac{2}{3})^2$

Recall that if the - symbol is not within the parentheses, it is not part of the base.

 $-(\frac{2}{3})^2=-(\frac{2}{3})\cdot(\frac{2}{3})$ Since the exponent is 2, write the base, $\frac{2}{3}$, as a factor 2 times. $=-\frac{2\cdot2}{3\cdot3}$ Multiply the numerators Multiply the denominators $=-\frac{4}{9}$

We read $-(\frac{2}{3})^2$ as "the opposite of two-thirds squared"