How does squaring a surd expression like $(k\sqrt{m})^2$ affect its components?

Understand the Problem

The question asks about the effect of squaring a surd expression in the form of $(k\sqrt{m})^2$. It wants to know if squaring only affects the integer coefficient 'k', only the surd part '\sqrt{m}', leaves the expression unchanged, or affects both the integer coefficient and the surd part.

Answer

Squaring $(k\sqrt{m})$ affects both $k$ and $\sqrt{m}$, resulting in $k^2m$.

Steps to Solve

Recall the properties of exponents

When squaring a term, we are multiplying it by itself. In this case, we have $(k\sqrt{m})^2 = (k\sqrt{m}) \cdot (k\sqrt{m})$.

Apply the commutative and associative properties of multiplication

We can rearrange the terms as follows: $(k\sqrt{m}) \cdot (k\sqrt{m}) = k \cdot k \cdot \sqrt{m} \cdot \sqrt{m}$.

Simplify the expression

$k \cdot k = k^2$ and $\sqrt{m} \cdot \sqrt{m} = m$. Therefore, the expression becomes: $k^2 \cdot m = k^2m$.

Analyze the result

The original expression $(k\sqrt{m})^2$ simplifies to $k^2m$. This shows that the squaring operation affects both the integer coefficient $k$ and the surd part $\sqrt{m}$. The integer coefficient $k$ is squared ($k^2$), and the surd part $\sqrt{m}$ is effectively removed, leaving just $m$.

Squaring the expression $(k\sqrt{m})$ affects both the integer coefficient and the surd part. The result is $k^2m$.

More Information

When squaring a surd expression like $(k\sqrt{m})^2$, you are essentially applying the exponent to both the coefficient $k$ and the square root $\sqrt{m}$. Squaring the coefficient $k$ results in $k^2$, while squaring the square root $\sqrt{m}$ results in $m$, thus eliminating the surd.

Tips

A common mistake is to only square the integer coefficient $k$ and forget to address the surd part $\sqrt{m}$. Another mistake would be to incorrectly square the surd part. Remember that $(\sqrt{m})^2 = m$.

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