In Exercises 21–32, simplify by factoring._____³√250x³

Verified step by step guidance

1. Step 1: Identify the expression to simplify. The expression is \( \sqrt[3]{250x^3} \).
2. Step 2: Break down the number 250 into its prime factors. 250 can be factored as \( 2 \times 5^3 \).
3. Step 3: Rewrite the expression using the prime factorization. The expression becomes \( \sqrt[3]{2 \times 5^3 \times x^3} \).
4. Step 4: Apply the property of cube roots that allows you to separate the factors: \( \sqrt[3]{2} \times \sqrt[3]{5^3} \times \sqrt[3]{x^3} \).
5. Step 5: Simplify each cube root. \( \sqrt[3]{5^3} = 5 \) and \( \sqrt[3]{x^3} = x \). Combine these with \( \sqrt[3]{2} \) to get the simplified expression.

Key Concepts

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. In algebra, this often involves identifying common factors or applying specific techniques such as grouping or using special products. Understanding how to factor is essential for simplifying expressions and solving equations.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. In this context, the cube root (³√) indicates the value that, when multiplied by itself three times, gives the original number. Simplifying radical expressions often requires factoring the radicand (the number inside the root) to identify perfect cubes or other factors that can be simplified.

Exponents and Powers

Exponents represent repeated multiplication of a base number. In the expression 250x³, the exponent 3 indicates that x is multiplied by itself three times. Understanding how to manipulate exponents, including the rules for multiplying and dividing powers, is crucial for simplifying expressions that involve both numerical coefficients and variable terms.