Problem 585 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\left(r^{16} s^{10}\right)^{\frac{1}{2}}\) (b) \(\left(u^{10} v^{5}\right)^{\frac{4}{5}}\)

Step-by-Step Solution

Answer

(a) \ r^8 s^5 \, (b) \ u^8 v^4

Step 1: Understand the Problem

The goal is to simplify the given expressions using the properties of exponents.

Step 2: Simplify Part (a)

The expression is \(\left( r^{16} s^{10} \right)^{\frac{1}{2}} \). Apply the power of a power property, which states \( (a^m)^n = a^{m \cdot n} \). This means: \(( r^{16} )^{ \frac{1}{2} } = r^{16 \cdot \frac{1}{2}} = r^8 \) and \( ( s^{10} )^{ \frac{1}{2} } = s^{10 \cdot \frac{1}{2}} = s^5 \). Therefore, the simplified expression is \( r^8 s^5 \).

Step 3: Simplify Part (b)

The expression is \(\left( u^{10} v^{5} \right)^{ \frac{4}{5} } \). Apply the power of a power property: \( ( u^{10} )^{\frac{4}{5}} = u^{10 \cdot \frac{4}{5}} = u^8 \) and \( ( v^{5} )^{ \frac{4}{5} } = v^{5 \cdot \frac{4}{5}} = v^4 \). Therefore, the simplified expression is \( u^8 v^4 \).

Key Concepts

properties of exponents

Understanding the properties of exponents is crucial when simplifying expressions. Exponents are essentially shorthand for repeated multiplication. Here are some key properties to remember:

1. **Product of Powers Property**: When multiplying two powers that have the same base, add their exponents: \( a^m \times a^n = a^{m+n} \).
2. **Quotient of Powers Property**: When dividing two powers that have the same base, subtract the exponent of the denominator from the exponent of the numerator: \( \frac{a^m}{a^n} = a^{m-n} \), given that \( a eq 0 \).
3. **Power of a Power Property**: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{m \times n} \).
4. **Power of a Product Property**: When raising a product to a power, raise each factor in the product to the power: \( (ab)^m = a^m b^m \).

These rules help in breaking down and simplifying complex exponent expressions.

simplifying expressions

Simplifying expressions involves using algebraic rules to rewrite expressions in their simplest form. The use of exponent properties can make this process easier. Let's see this with the given examples.

In Part (a), the expression to simplify is \( \big( r^{16} s^{10} \big)^{\frac{1}{2}} \). By applying the power of a power property, we simplify this to: \( (r^{16})^{\frac{1}{2}} \times (s^{10})^{\frac{1}{2}} \).

Using the power of a power rule here gives \( r^{16 \times \frac{1}{2}} \times s^{10 \times \frac{1}{2}} \). Simplifying the exponents results in \( r^8 \times s^5 \).

In Part (b), the expression is \( \big( u^{10} v^{5} \big)^{\frac{4}{5}} \). Similarly, applying the power of a power property, we get: \( (u^{10})^{\frac{4}{5}} \times (v^{5})^{\frac{4}{5}} \).

This simplifies to \( u^{10 \times \frac{4}{5}} \times v^{5 \times \frac{4}{5}} \), giving \( u^8 \times v^4 \). This breakdown of each step clarifies the process.

power of a power rule

The power of a power rule is one of the most important tools in simplifying expressions involving exponents. This rule states that when you have an exponent raised to another exponent, you multiply the exponents: \( (a^m)^n = a^{m \times n} \). This rule allows you to collapse more complex exponentiation into a simpler form.

For instance, in the context of our exercises:

* For Part (a), \( (r^{16})^{\frac{1}{2}} = r^{16 \times \frac{1}{2}} = r^8 \) and similarly \( (s^{10})^{\frac{1}{2}} = s^{10 \times \frac{1}{2}} = s^5 \).

* For Part (b), \( (u^{10})^{\frac{4}{5}} = u^{10 \times \frac{4}{5}} = u^8 \) and \( (v^{5})^{\frac{4}{5}} = v^{5 \times \frac{4}{5}} = v^4 \).

By employing the power of a power rule, you can greatly simplify expressions and solve problems more efficiently. Always start by applying this rule to see if the expression can be made simpler.