Answered

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What is the simplest form of [tex]\sqrt{a^7}[/tex]?

A. [tex]a^2 \sqrt{a}[/tex]
B. [tex]a^3 \sqrt{a}[/tex]
C. [tex]a^3 \sqrt{a^2}[/tex]
D. [tex]3 a \sqrt{a}[/tex]

Sagot :

GinnyAnswer
To find the simplest form of \(\sqrt{a^7}\), let's break down the exponent rules and manipulation of square roots.

First, recall that the square root of a power can be expressed as:

[tex]\[ \sqrt{a^b} = a^{b/2} \][/tex]

Given the expression \(\sqrt{a^7}\), we can apply this rule:

[tex]\[ \sqrt{a^7} = a^{7/2} \][/tex]

Next, express \(a^{7/2}\) in terms of an integer power times a square root. Notice that:

[tex]\[ a^{7/2} = a^{3.5} = a^{3+0.5} = a^3 \cdot a^{0.5} \][/tex]

Since \(a^{0.5} = \sqrt{a}\), we can rewrite it as:

[tex]\[ a^{3+0.5} = a^3 \cdot \sqrt{a} \][/tex]

So, the simplest form of \(\sqrt{a^7}\) is:

[tex]\[ a^3 \cdot \sqrt{a} \][/tex]

Thus, the correct answer is:
[tex]\[ a^3 \sqrt{a} \][/tex]
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