Q. \((x^2)^2\)
Answer
Simplify \(\left(x^2\right)^2\) by using the power rule:
\[
\left(x^2\right)^2 = x^{2\cdot 2} = x^4.
\]
Final result: \(x^4\)
Detailed Explanation
We want to simplify the expression \( (x^{2})^{2} \).
Step 1: Identify the exponent rule
When you have a power raised to another power, you can use the rule:
\[ (a^{m})^{n} = a^{m n} \]
Step 2: Match the parts
In \( (x^{2})^{2} \), the base is \(x\). The outer exponent is \(2\). The inner exponent is also \(2\).
So we can treat \( (x^{2})^{2} \) as \( (x^{m})^{n} \) with \(m=2\) and \(n=2\).
Step 3: Multiply the exponents
Using the rule \( (a^{m})^{n} = a^{m n} \), we get:
\[ (x^{2})^{2} = x^{2 \cdot 2} \]
Step 4: Compute the product
\[ x^{2 \cdot 2} = x^{4} \]
Final Answer
\[ (x^{2})^{2} = x^{4} \]
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Algebra FAQ
Evaluate \( (x^2)^2 \).
\( (x^2)^2 = x^{2\cdot2} = x^4 \).
Can you simplify \( (x^2)^2 \) using exponent rules?
Yes. Use \( (a^m)^n = a^{mn} \). Here \( a=x \), \( m=2 \), \( n=2 \), so \( x^{4} \).
What is \( (x^2)^3 \) and how is it similar?
\( (x^2)^3 = x^{2\cdot3} = x^6 \). Same rule: multiply exponents.
How do you expand \( (x^2)^2 \) without exponent rules?
\( (x^2)^2 = (x^2)(x^2) = x^{2+2} = x^4 \).
Is \( (x^2)^2 \) equal to \( x^2 \cdot x^2 \)?
Yes. \( (x^2)^2 = x^2 \cdot x^2 \). Then combine to get \( x^4 \).
Does the simplification change if \( x \) is negative?
No. Since squaring removes the sign twice, \( (x^2)^2 = x^4 \) for all real \( x \).
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