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Simplify.
Remove all perfect squares from inside the square roots. Assume
a and
b are positive.

sqrt(42a^(4)b^(6))=◻

Simplify. $\newline$ Remove all perfect squares from inside the square roots. Assume $a$ and $b$ are positive. $\newline$ $\sqrt{42 a^{4} b^{6}}=\square$

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Simplify radical expressions with variables

Full solution

Q. Simplify.

\newline

Remove all perfect squares from inside the square roots. Assume

a

and

b

are positive.

\newline

\sqrt{42 a^{4} b^{6}}=\square

Factorize Expression: Factorize the expression inside the square root to identify perfect squares. $\newline$ The expression inside the square root is $42a^{4}b^{6}$ . We can factorize $42$ into its prime factors and express $a^{4}$ and $b^{6}$ as squares of numbers. $\newline$ $42 = 2 \times 3 \times 7$ $\newline$ $a^{4} = (a^2)^2$ $\newline$ $b^{6} = (b^3)^2$
Rewrite with Factorization: Rewrite the expression inside the square root using the factorization. $\newline$ Now we can rewrite the expression inside the square root as: $\newline$ $\sqrt{42a^{4}b^{6}} = \sqrt{2 \times 3 \times 7 \times (a^{2})^{2} \times (b^{3})^{2}}$
Separate Perfect Squares: Separate the perfect squares from the non-perfect squares inside the square root. We can separate the perfect squares $(a^2)^2$ and $(b^3)^2$ from the non-perfect squares $2$ , $3$ , and $7$ . $\sqrt{42a^{4}b^{6}} = \sqrt{(a^2)^2 \cdot (b^3)^2 \cdot 2 \cdot 3 \cdot 7}$
Take Out Perfect Squares: Take the perfect squares out of the square root. Since the square root of a square is the number itself, we can take $a^2$ and $b^3$ out of the square root. $\sqrt{42a^{4}b^{6}} = a^2 \times b^3 \times \sqrt{2 \times 3 \times 7}$
Simplify Under Square Root: Simplify the expression under the square root. $\newline$ The expression under the square root cannot be simplified further since $2$ , $3$ , and $7$ are all prime numbers. $\newline$ Therefore, the final simplified expression is: $\newline$ $\sqrt{42a^{4}b^{6}} = a^{2} \times b^{3} \times \sqrt{2 \times 3 \times 7}$

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