Problem 48 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(q^{\frac{3}{2}} \sqrt{63 p}\)

Step-by-Step Solution

Answer

The simplified expression is \(3q^{\frac{3}{2}} \sqrt{7p}\).

Step 1: Simplify the Square Root

The expression inside the square root is \( \sqrt{63p} \). Let's simplify this by factoring out the square and any perfect squares in 63: \( 63 = 9 \times 7 \), and \( \sqrt{9} = 3 \). So, \( \sqrt{63p} = \sqrt{9 \times 7p} = 3\sqrt{7p} \).

Step 2: Recombine the Expression

Now, combine this with the \( q^{\frac{3}{2}} \): \( q^{\frac{3}{2}} \sqrt{63p} = q^{\frac{3}{2}} \cdot 3\sqrt{7p} \).

Step 3: Simplify the Coefficients

We can simplify the expression by multiplying the coefficients and leaving the variable expressions separate: \( 3q^{\frac{3}{2}} \sqrt{7p} \).

Key Concepts

Square Roots

When working with square roots, you're dealing with numbers or expressions raised to the power of 1/2. Square roots help us find a number which, when multiplied by itself, gives the original number. In algebra, simplifying square roots often requires breaking them down into factors.

To simplify, identify pairs of factors from the original number inside the root. Take out any perfect square factors, as these can be written as base values outside of the square root.

• For example, consider \( \sqrt{63} \): Since \( 63 = 9 \times 7 \), and \( \sqrt{9} = 3 \), we have \( \sqrt{63} = 3\sqrt{7} \).
• This means 9 is a perfect square factor, simplifying our task of reducing the expression.

Remember, the same technique applies to any expression inside the square root that also has variables, like \( \sqrt{63p} \). Just separate numbers and variables into their factor pairs.

Exponents

Exponents are all about repeated multiplication. They show how many times a number, known as the base, is multiplied by itself. Fractional exponents, like \( q^{\frac{3}{2}} \), are an extension of this concept—combining power and root operations.

The fractional exponent \( \frac{3}{2} \) represents a base raised to the third power and taking the square root of the result. Mathematically, this can be expressed in parts:

• First, raise the base to the power of 3, \( q^3 \).
• Second, apply the square root: \( \sqrt{q^3} \).

This is often used to simplify expressions in algebra. Understanding how to manipulate exponents helps in recombining terms and simplifying further, as seen when multiplying coefficients or combining expressions like \( q^{\frac{3}{2}} \) with other parts.

Factoring

Factoring is the process of breaking down numbers or expressions into their multiplicative components. This algebraic technique is key to simplifying complex expressions.

Start by identifying numbers and variables that can be divided into smaller parts or factors. Utilize the concept of prime factorization where you express numbers as a product of primes.

• For instance, \( 63 = 3 \times 3 \times 7 \), which makes the factoring \( 9 \times 7 \), where 9 is a perfect square.
• In the expression \( \sqrt{63p} \), we broke it down to \( 3\sqrt{7p} \), neatly separating constants from radicals and simplifying the square root term.

By practicing factoring, you'll find it easier to work with more complex algebraic expressions. It also makes expressions clearer and manageable, which is vital when you’re aiming to simplify or solve equations.