Select the correct answer.
Which expression is equivalent to $4 x^2 \sqrt{5 x^4} \cdot 3 \sqrt{5 x^8}$, if $x \neq 0$ ?
A. $12 x^{10} \sqrt{5}$
B. $60 x^8$
C. $35 x^{18}$
D. $7 x^{10} \sqrt{5}$
Answer (2)
The expression 4 x 2 5 x 4 ⋅ 3 5 x 8 simplifies to 60 x 8 . Therefore, the correct answer is option B: 60 x 8 .
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Rewrite the expression using exponent rules.
Simplify the expression by distributing the exponents.
Combine the terms by multiplying the coefficients and adding the exponents of x .
Simplify the expression to get the final answer: 60 x 8 .
Explanation
Understanding the Problem We are asked to find an expression equivalent to 4 x 2 5 x 4 ⋅ 3 5 x 8 , given that x = 0 . We will simplify the expression using exponent rules and properties of square roots.
Rewriting with Exponents First, rewrite the expression using exponent rules: 4 x 2 5 x 4 ⋅ 3 5 x 8 = 4 x 2 ( 5 x 4 ) 1/2 ⋅ 3 ( 5 x 8 ) 1/2
Simplifying the Expression Next, simplify the expression by distributing the exponents: 4 x 2 ( 5 x 4 ) 1/2 ⋅ 3 ( 5 x 8 ) 1/2 = 4 x 2 ⋅ 5 x 2 ⋅ 3 ⋅ 5 x 4
Combining Terms Now, combine the terms by multiplying the coefficients and adding the exponents of x : 4 ⋅ 3 ⋅ 5 ⋅ 5 ⋅ x 2 ⋅ x 2 ⋅ x 4 = 12 ⋅ 5 ⋅ x 2 + 2 + 4
Final Simplification Finally, simplify the expression: 12 ⋅ 5 ⋅ x 2 + 2 + 4 = 60 x 8
Conclusion Therefore, the expression equivalent to 4 x 2 5 x 4 ⋅ 3 5 x 8 is 60 x 8 .
Examples
Understanding how to simplify expressions with radicals and exponents is crucial in many areas, such as physics and engineering. For instance, when calculating the energy of a particle or analyzing the behavior of waves, you often encounter expressions involving radicals and exponents. Simplifying these expressions allows for easier calculations and a better understanding of the underlying phenomena. This skill is also fundamental in computer graphics when dealing with transformations and scaling of objects, where efficient calculations are essential for smooth rendering.
