Simplify $\sqrt{25x^7} \times \sqrt{8x^3}$
Understand the Problem
The question asks to simplify the product of two square roots: $\sqrt{25x^7}$ and $\sqrt{8x^3}$. The user needs to simplify each square root and combine like terms.
Answer
$10x^5\sqrt{2}$
Answer for screen readers
$10x^5\sqrt{2}$
Steps to Solve
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Simplify $\sqrt{25x^7}$
Rewrite $x^7$ as $x^6 \cdot x$ to extract the perfect square.
$\sqrt{25x^7} = \sqrt{25 \cdot x^6 \cdot x} = \sqrt{25} \cdot \sqrt{x^6} \cdot \sqrt{x} = 5x^3\sqrt{x}$
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Simplify $\sqrt{8x^3}$
Rewrite $8$ as $4 \cdot 2$ and $x^3$ as $x^2 \cdot x$ to extract the perfect squares.
$\sqrt{8x^3} = \sqrt{4 \cdot 2 \cdot x^2 \cdot x} = \sqrt{4} \cdot \sqrt{x^2} \cdot \sqrt{2x} = 2x\sqrt{2x}$
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Multiply the simplified expressions
Multiply the results from the previous steps:
$5x^3\sqrt{x} \cdot 2x\sqrt{2x} = 5 \cdot 2 \cdot x^3 \cdot x \cdot \sqrt{x} \cdot \sqrt{2x} = 10x^4\sqrt{2x^2}$
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Simplify the resulting square root
Simplify $\sqrt{2x^2}$ as $\sqrt{x^2} \cdot \sqrt{2}$.
$10x^4\sqrt{2x^2} = 10x^4 \cdot x \sqrt{2} = 10x^5\sqrt{2}$
$10x^5\sqrt{2}$
More Information
The simplified form of $\sqrt{25x^7} \times \sqrt{8x^3}$ is $10x^5\sqrt{2}$.
Tips
A common mistake is not simplifying the radicals completely, such as leaving $\sqrt{8}$ as is instead of simplifying it to $2\sqrt{2}$. Another frequent error is incorrectly combining the exponents of $x$. Remember to add the exponents when multiplying terms with the same base.
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