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Which expression is equivalent to [tex]\frac{\left(2 g^5\right)^3}{\left(4 h^2\right)^3}[/tex]?

A. [tex]\frac{g^{15}}{8 h^6}[/tex]
B. [tex]\frac{g^5}{2 h^2}[/tex]
C. [tex]\frac{g^{15}}{2 h^6}[/tex]
D. [tex]\frac{g^8}{8 h^5}[/tex]

Sagot :

To determine which expression is equivalent to [tex]\(\frac{(2 g^5)^3}{(4 h^2)^3}\)[/tex], we can simplify it step by step:

1. Simplify the numerator: [tex]\((2 g^5)^3\)[/tex]

- Raise the constant 2 to the power of 3: [tex]\(2^3 = 8\)[/tex].
- Raise [tex]\(g^5\)[/tex] to the power of 3: [tex]\((g^5)^3 = g^{5 \cdot 3} = g^{15}\)[/tex].

So, the numerator [tex]\((2 g^5)^3\)[/tex] simplifies to [tex]\(8 g^{15}\)[/tex].

2. Simplify the denominator: [tex]\((4 h^2)^3\)[/tex]

- Express the 4 as [tex]\(2^2\)[/tex], so rewrite [tex]\(4 h^2\)[/tex] as [tex]\((2^2 h^2)\)[/tex].
- Raise [tex]\(2^2\)[/tex] to the power of 3: [tex]\((2^2)^3 = 2^{2 \cdot 3} = 2^6 = 64\)[/tex].
- Raise [tex]\(h^2\)[/tex] to the power of 3: [tex]\((h^2)^3 = h^{2 \cdot 3} = h^6\)[/tex].

So, the denominator [tex]\((4 h^2)^3\)[/tex] simplifies to [tex]\(64 h^6\)[/tex].

3. Combine the simplified numerator and denominator:

[tex]\[ \frac{(2 g^5)^3}{(4 h^2)^3} = \frac{8 g^{15}}{64 h^6} \][/tex]

4. Simplify the fraction:

- Simplify the constant fraction: [tex]\(\frac{8}{64} = \frac{1}{8}\)[/tex].

Thus, the simplified expression is:

[tex]\[ \frac{8 g^{15}}{64 h^6} = \frac{1}{8} \cdot \frac{g^{15}}{h^6} = \frac{g^{15}}{8 h^6} \][/tex]

Therefore, the equivalent expression is [tex]\(\frac{g^{15}}{8 h^6}\)[/tex], which matches the first choice:

[tex]\[\frac{g^{15}}{8 h^6}\][/tex]