Problem 80 - College Algebra with Modeling and Visualization

Question

Simplify the expression and write it with rational exponents. Assume that all variables are positive. $$ \left(y^{10} z^{4}\right)^{1 / 4} $$

Step-by-Step Solution

Answer

The simplified expression is \(y^{2.5} z^{1}\).

Step 1: Identify the expression

The given expression is \( (y^{10} z^{4})^{1/4} \). Our goal is to simplify the expression and express it using rational exponents.

Step 2: Apply the exponent rule

Use the exponent rule where \((a^m)^n = a^{m imes n}\) to simplify the given expression. This means we apply the power of \(1/4\) to each variable inside the parentheses: \( (y^{10})^{1/4} \times (z^4)^{1/4} \).

Step 3: Simplify each term

Simplify each part individually using the rule from step 2:1. For \((y^{10})^{1/4}\), compute \(10 \times 1/4 = 10/4 = 2.5\), so \((y^{10})^{1/4} = y^{2.5}\).2. For \((z^{4})^{1/4}\), compute \(4 \times 1/4 = 4/4 = 1\), so \((z^{4})^{1/4} = z^{1}\).

Step 4: Combine the results

Combine the simplified forms from step 3: \(y^{2.5} \cdot z^{1}\). This is already using rational exponents.

Key Concepts

Exponent Rules

Understanding exponent rules is essential when dealing with expressions involving exponents. An exponent indicates how many times a number, known as the base, is multiplied by itself. For example, in the expression \( a^m \), \( m \) is the exponent and \( a \) is the base. Learning these rules helps simplify complex expressions.
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One key exponent rule is the power of a power rule, which states \((a^m)^n = a^{m \times n}\). This rule helps simplify expressions where an exponent is applied to another exponent. Rather than multiplying the base multiple times in a row, you can multiply the exponents directly and simplify the expression in just one step.
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In the original exercise, we applied this rule by taking \((y^{10} z^{4})^{1/4}\) and using the power of a power rule. We processed each variable separately by multiplying the internal exponent by the external exponent, easing the problem into manageable pieces.

Simplifying Expressions

Simplifying expressions involves rewriting them in a form that's easier to understand or use. The main goal is to make mathematical operations simpler and reduce the possibility of making errors.
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When an expression includes multiple variables raised to various powers, such as \((y^{10} z^{4})^{1/4}\), simplification becomes crucial. Here, simplifying means applying exponent rules correctly and reducing the expression to terms you can easily handle. This may involve multiplying or dividing powers according to the rules, and then writing it in an elegant form.
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In our example, each part of the expression, \((y^{10})^{1/4}\) and \((z^4)^{1/4}\), was dealt with separately, and simplified into its simplest rational exponent form, \(y^{2.5}\) and \(z^1\). By doing this, the expression is easier to manipulate in further calculations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operators. They can represent real-world situations or purely abstract scenarios. Variables, like \(y\) and \(z\) in our example, make algebra flexible and powerful.
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When working with algebraic expressions, it's important to know their structure and how to manipulate them through operations like addition, subtraction, multiplication, and exponentiation. Rational exponents, as shown in our exercise, are just one part of this web of operations that helps express complex ideas in simple forms.
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In the expression \((y^{10} z^{4})^{1/4}\), the use of rational exponents provides a way to express roots and fractional powers. This is vital in both simplification and solving equations, as it creates a bridge between roots and powers, offering flexibility and precision.