√(16x) • √(16x^5)
Understand the Problem
The question is asking us to simplify the expression involving square roots. Specifically, it consists of multiplying two square roots together: √(16x) and √(16x^5).
Answer
$16x^3$
Answer for screen readers
The simplified expression is $16x^3$.
Steps to Solve
- Combine the Square Roots We can use the property of square roots that states $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$. Therefore, we can combine our two square roots:
$$ \sqrt{16x} \cdot \sqrt{16x^5} = \sqrt{(16x) \cdot (16x^5)} $$
- Multiply the Terms Inside the Square Root Next, we can multiply the expressions inside the square root:
$$ (16x) \cdot (16x^5) = 16^2 \cdot x^{1+5} = 256x^6 $$
- Finish Simplifying the Square Root Now that we have a single expression inside the square root, we simplify it:
$$ \sqrt{256x^6} $$
- Extract Square Roots Now we can extract the square root:
$$ \sqrt{256} = 16 \quad \text{and} \quad \sqrt{x^6} = x^3 $$
Combining these results gives us:
$$ \sqrt{256x^6} = 16x^3 $$
The simplified expression is $16x^3$.
More Information
The square root of a product can often be simplified by first multiplying the terms inside the square root. In this case, both the numeric and variable components were straightforward to simplify.
Tips
- Not Combining the Roots Correctly: Some might forget to combine the square roots correctly before simplifying.
- Incorrectly Simplifying Exponents: Ensure that you add exponents correctly when multiplying terms: $x^1 \cdot x^5$ should equal $x^{1+5}=x^6$.
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