Question

which two expressions are equivalent to each other?
$(2^3)^4$
$\frac{2^{24}}{2^2}$
$(2^2)^6$
$\frac{1}{2^{12}}$

Explanation:

Step1: Simplify \((2^{3})^{4}\)

Using the power - of - a - power rule \((a^{m})^{n}=a^{m\times n}\), for \((2^{3})^{4}\), we have \(2^{3\times4}=2^{12}\).

Step2: Simplify \(\frac{2^{24}}{2^{2}}\)

Using the quotient rule of exponents \(\frac{a^{m}}{a^{n}}=a^{m - n}\), for \(\frac{2^{24}}{2^{2}}\), we get \(2^{24-2}=2^{22}\).

Step3: Simplify \((2^{2})^{6}\)

Using the power - of - a - power rule \((a^{m})^{n}=a^{m\times n}\), for \((2^{2})^{6}\), we have \(2^{2\times6}=2^{12}\).

Step4: Simplify \(\frac{1}{2^{12}}\)

We know that \(\frac{1}{a^{n}}=a^{-n}\), so \(\frac{1}{2^{12}} = 2^{-12}\).

Now we compare the simplified forms:

• \((2^{3})^{4}=2^{12}\)
• \(\frac{2^{24}}{2^{2}}=2^{22}\)
• \((2^{2})^{6}=2^{12}\)
• \(\frac{1}{2^{12}}=2^{-12}\)

So \((2^{3})^{4}\) and \((2^{2})^{6}\) are equivalent because both simplify to \(2^{12}\).

Answer:

\(\boldsymbol{(2^{3})^{4}}\) and \(\boldsymbol{(2^{2})^{6}}\)