Q. \((x^2)^3\)
Answer
Use the power of a power rule: \((a^m)^n=a^{mn}\). Here \((x^2)^3=x^{2\cdot 3}=x^6\).
\[
(x^2)^3 = x^6
\]
Detailed Explanation
We want to evaluate the expression \( (x^2)^3 \).
Step 1: Identify the exponent rules.
We use the power of a power rule:
\[
(a^m)^n = a^{m n}
\]
Here, \(a = x^2\), \(m = 2\), and \(n = 3\).
Step 2: Apply the power of a power rule.
Rewrite \( (x^2)^3 \) as:
\[
(x^2)^3 = x^{2 \cdot 3}
\]
So we multiply the exponents \(2\) and \(3\).
Step 3: Compute the product of the exponents.
\[
2 \cdot 3 = 6
\]
Therefore, the expression becomes:
\[
(x^2)^3 = x^6
\]
Final Answer:
\[
(x^2)^3 = x^6
\]
See full solution
Algebra FAQ
How do you simplify \((x^2)^3\)?
Use the power of a power rule: \((a^m)^n=a^{mn}\). So \((x^2)^3=x^{2\cdot 3}=x^6\).
What is the exponent when \((x^2)^3\) is rewritten as a single power of \(x\)?
Multiply exponents: \(2\cdot 3=6\). Therefore, \((x^2)^3=x^6\).
Why isn’t \(\left(x^2\right)^3\) equal to \(x^5\) or \(x^{2^3}=x^3\)?
Exponent rules apply: \(2\) is multiplied by \(3\), not added or replaced. Correct exponent is \(2\cdot 3=6\), so \(x^6\).
Expand \((x^2)^3\) without exponent rules first, then simplify.
\((x^2)^3=x^2\cdot x^2\cdot x^2\). Multiply: \(x^{2+2+2}=x^6\).
Does the simplification depend on whether \(x\) is positive or negative?
No. For integer exponents, exponent rules still hold. \((x^2)^3=x^6\) for any real \(x\).
What common rule matches \((x^2)^3\) in \((a^m)^n\) form?
The power of a power rule: \((a^m)^n=a^{mn}\). Here, \(a=x\), \(m=2\), \(n=3\), giving \(x^{6}\).
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