Problem 22 - Precalculus Mathematics for Calculus

Question

Evaluate each expression. (a) \(\sqrt{7} \sqrt{28}\) (b) \(\frac{\sqrt{48}}{\sqrt{3}}\) (c) \(\sqrt[4]{24} \sqrt[4]{54}\)

Step-by-Step Solution

Answer

(a) 14, (b) 4, (c) 6.

Step 1: Simplify Square Roots for Part (a)

To evaluate \(\sqrt{7} \sqrt{28}\), first simplify the square roots. Note that \(28 = 4 \times 7\) and \(\sqrt{28} = \sqrt{4 \times 7} = \sqrt{4} \sqrt{7} = 2 \sqrt{7}\). Thus, the given expression becomes \(\sqrt{7} \times 2 \sqrt{7}\).

Step 2: Combine and Simplify the Expression

Combine the terms from Step 1, \(\sqrt{7} \times 2 \sqrt{7} = 2 \times \sqrt{7} \times \sqrt{7}\). Since \(\sqrt{7} \times \sqrt{7} = 7\), the expression simplifies to \(2 \times 7 = 14\).

Step 3: Simplify the Expression for Part (b)

Evaluate \(\frac{\sqrt{48}}{\sqrt{3}}\). Begin by simplifying \(\sqrt{48}\): note that \(48 = 16 \times 3\), so \(\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \sqrt{3} = 4 \sqrt{3}\). Thus, the expression becomes \(\frac{4 \sqrt{3}}{\sqrt{3}}\).

Step 4: Cancel Out Like Terms

Since \(\frac{4 \sqrt{3}}{\sqrt{3}} = 4\), the \(\sqrt{3}\) terms cancel each other out, leaving the simplified expression as 4.

Step 5: Simplify Fourth Roots for Part (c)

To solve \(\sqrt[4]{24} \sqrt[4]{54}\), notice that \(24 = 2^3 \times 3\) and \(54 = 2 \times 3^3\). Multiply the numbers under the root: \(24 \times 54 = 2^4 \times 3^4\). Therefore, \(\sqrt[4]{24} \sqrt[4]{54} = \sqrt[4]{2^4 \times 3^4}\).

Step 6: Evaluate the Fourth Root

Since \(\sqrt[4]{x^4} = x\), \(\sqrt[4]{2^4 \times 3^4} = \sqrt[4]{(2 \times 3)^4} = 2 \times 3 = 6\). Hence, the expression simplifies to 6.

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics, often introduced in pre-calculus courses. They allow us to determine a number that, when multiplied by itself, results in the original value under the root. The symbol for a square root is \(\sqrt{}\) and can be used, for example, as \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).

When working with square roots in expressions, it is usually helpful to simplify them by finding perfect square factors. For instance, \(\sqrt{28}\) can be simplified to \(\sqrt{4 \times 7} = \sqrt{4} \cdot \sqrt{7} = 2\sqrt{7}\), since 4 is a perfect square (as \(2 \times 2 = 4\).

• Look for largest perfect square factor under the root.
• Break down the expression into individual square roots.
• Simplify using the perfect square values.

By simplifying, calculations become easier and results more precise.

Fourth Roots

The fourth root is a less common operation compared to the square root, but it holds a similar purpose. It helps us find a number that, when raised to the fourth power, results in the original value under the root. The fourth root is denoted as \(\sqrt[4]{}\). For instance, \(\sqrt[4]{81} = 3\) because \(3^4 = 81\).

To simplify fourth roots, look for factors under the root that are raised to the power of four. Consider \(\sqrt[4]{24}\) and \(\sqrt[4]{54}\) given in our exercise. Here, we first express each factor in terms of primes and then combine them:

• Express the number as a product of powers of primes. For example, \(24 = 2^3 \times 3\).
• Multiply expressions under the roots before simplifying.
• Apply \(\sqrt[4]{a^4}\) to simplify directly. This means if you have \(\sqrt[4]{x^4}\), it directly simplifies to \(x\).

This allows us to simplify complicated expressions efficiently, delivering precise results.

Simplifying Expressions

Simplifying expressions, especially those involving roots, is crucial to solving mathematical problems efficiently. Simplification makes complex calculations manageable and results easier to interpret.

To simplify an expression, follow these key steps:

• Identify and simplify roots individually and where possible, eliminate root symbols by solving for exact numbers.
• Factorize numbers under the roots into primes or perfect powers where applicable.
• Combine like terms and perform arithmetic operations as per standard mathematical rules.

In the exercise solution, simplifying \(\sqrt{7} \cdot \sqrt{28}\) led to breaking \(\sqrt{28}\) down to \(2\sqrt{7}\), allowing further simplification to \(14\). Similarly, simplifying \(\frac{\sqrt{48}}{\sqrt{3}}\) by cancelling out like terms led to a neat result of \(4\). Finally, even a complex operation such as \(\sqrt[4]{24} \cdot \sqrt[4]{54}\) is reduced to a simple number \(6\) through careful handling of roots.

Remember: simplifying isn't just about doing less work but doing it smarter!