Question
An expression is shown.
(2/3)^4 x (2/3)^-6
Which of the following is equivalent to the expression shown?
Responses
A
the fraction with numerator 2 squared and denominator 3 squared
B
the fraction with numerator 3 squared and denominator 2 squared
C
the fraction with numerator 3 to the 24th power and denominator 2 to the 24th power
D
the fraction with numerator 3 squared and denominator 2
Answers
GPT-4o mini
To simplify the expression \((2/3)^4 \times (2/3)^{-6}\), we can use the property of exponents that states \(a^m \times a^n = a^{m+n}\).
First, we combine the exponents:
\[
(2/3)^4 \times (2/3)^{-6} = (2/3)^{4 + (-6)} = (2/3)^{-2}
\]
Now, an exponent of \(-2\) indicates that we can take the reciprocal and then square the base:
\[
(2/3)^{-2} = \frac{1}{(2/3)^2}
\]
Calculating \((2/3)^2\) gives:
\[
(2/3)^2 = \frac{2^2}{3^2} = \frac{4}{9}
\]
Thus, we find:
\[
(2/3)^{-2} = \frac{1}{(2/3)^2} = \frac{1}{\frac{4}{9}} = \frac{9}{4}
\]
Now, we can write \(\frac{9}{4}\) in terms of the original expression. This is the same as \(\frac{3^2}{2^2}\):
\[
\frac{9}{4} = \frac{3^2}{2^2}
\]
Now let's check the provided responses:
A: the fraction with numerator 2 squared and denominator 3 squared (incorrect)
B: the fraction with numerator 3 squared and denominator 2 squared (correct: \(\frac{3^2}{2^2}\))
C: the fraction with numerator 3 to the 24th power and denominator 2 to the 24th power (incorrect)
D: the fraction with numerator 3 squared and denominator 2 (incorrect)
The equivalent expression is:
**B: the fraction with numerator 3 squared and denominator 2 squared.**
First, we combine the exponents:
\[
(2/3)^4 \times (2/3)^{-6} = (2/3)^{4 + (-6)} = (2/3)^{-2}
\]
Now, an exponent of \(-2\) indicates that we can take the reciprocal and then square the base:
\[
(2/3)^{-2} = \frac{1}{(2/3)^2}
\]
Calculating \((2/3)^2\) gives:
\[
(2/3)^2 = \frac{2^2}{3^2} = \frac{4}{9}
\]
Thus, we find:
\[
(2/3)^{-2} = \frac{1}{(2/3)^2} = \frac{1}{\frac{4}{9}} = \frac{9}{4}
\]
Now, we can write \(\frac{9}{4}\) in terms of the original expression. This is the same as \(\frac{3^2}{2^2}\):
\[
\frac{9}{4} = \frac{3^2}{2^2}
\]
Now let's check the provided responses:
A: the fraction with numerator 2 squared and denominator 3 squared (incorrect)
B: the fraction with numerator 3 squared and denominator 2 squared (correct: \(\frac{3^2}{2^2}\))
C: the fraction with numerator 3 to the 24th power and denominator 2 to the 24th power (incorrect)
D: the fraction with numerator 3 squared and denominator 2 (incorrect)
The equivalent expression is:
**B: the fraction with numerator 3 squared and denominator 2 squared.**