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Chapter 8: Problem 95

Simplify each radical. Assume that \(x \geq 0\) $$ \sqrt[4]{48^{2}} $$

Short Answer

Expert verified
4\sqrt{3}

Step by step solution

01

Rewrite the expression with exponents

Start by expressing the given radical using exponents. The radical \( \sqrt[4]{48^{2}} \ \) can be written as \( 48^{2/4} \ \).
02

Simplify the exponent

Next, simplify the exponent \( \frac{2}{4} \ \) to its lowest terms. Divide both the numerator and the denominator by their greatest common divisor which is 2. \( \frac{2}{4} = \frac{1}{2} \ \).
03

Recalculate the expression

Replace the exponent with the simplified version: \( 48^{1/2} \ \). This expression is equivalent to \( \sqrt{48} \ \).
04

Simplify the square root

Simplify \( \sqrt{48} \ \) by breaking it down into its prime factors. \( 48 = 16 \cdot 3 \). Since \( 16 \) is a perfect square, \( \sqrt{48} = \sqrt{16 \cdot 3} = \sqrt{16} \cdot \sqrt{3} = 4\sqrt{3} \ \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponents
Understanding exponents is key to tackling radical expressions. Exponents, often noted as a small number above a base number, tell us how many times to multiply the base by itself. For instance, in the expression \( 48^{2} \), 48 is the base and 2 is the exponent. This means 48 multiplied by itself, or \( 48 \times 48 \).
  • When simplifying radicals, we often convert radicals to expressions with fractional exponents.
  • For example, \( \sqrt[4]{48^{2}} \) can be written as \( 48^{2/4} \).
By simplifying the fraction \( 2/4 \) to \( 1/2 \), we transform the original problem more simply: the square root of 48, written as \( 48^{1/2} \) or \( \sqrt{48} \). Understanding how to work with exponents helps reveal simpler forms of complex expressions.
Prime Factorization
Prime factorization helps simplify expressions, especially when dealing with radicals. It involves breaking down a number into its prime factors鈥攏umbers only divisible by 1 and themselves. For example, consider the number 48:
  • First, divide it by the smallest prime number possible: \( 48 \div 2 = 24 \).
  • Continue dividing until you break it down entirely into primes: \( 24 \div 2 = 12 \), \( 12 \div 2 = 6 \), \( 6 \div 2 = 3 \).
  • The prime factorization of 48 is \( 2^4 \times 3 \).
Prime factorization is useful when simplifying a radical. For instance, \( \sqrt{48} \) becomes easier to manage by recognizing \( 48 = 2^4 \times 3 \). Simplifying the radical then isolates perfect squares from non-square terms: \( \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \).
Radical Expressions
Radical expressions involve roots, like square roots or fourth roots, which are the opposite operation of exponents. For instance, \( \sqrt[4]{48^2} \) implies finding a number which, when raised to the fourth power, gives \( 48^2 \).
When simplifying a radical expression:
  • First convert it to an exponent form, if possible. \( \sqrt[4]{48^2} \) is rewritten as \( 48^{2/4} \).
  • Simplify the fractional exponent: \( \frac{2}{4} = \frac{1}{2} \).
  • It transforms the expression into \( 48^{1/2} \) or \( \sqrt{48} \).
  • Break down the radicand (the number inside the radical) into its prime factors.
  • Simplify the radical by pulling out pairs of prime factors. Since 48 = 16 x 3, \( \sqrt{48} \) becomes \( \sqrt{16 \times 3} = 4\sqrt{3} \).
These steps help simplify and understand radical expressions better, leading to exact or more manageable forms.

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Most popular questions from this chapter

Perform the indicated operations. Give answers in standard form. $$ \frac{3}{2-i}+\frac{5}{1+i} $$

Write with rational exponents, and then apply the properties of exponents. Assume that all radicands represent positive real numbers. Give answers in exponential form. $$ \sqrt[3]{\sqrt[5]{\sqrt{y}}} $$

Simplify each radical. Assume that \(x \geq 0\) $$ \sqrt[12]{x^{44}} $$

Find the distance between each pair of points. $$ (\sqrt{2}, \sqrt{6}) \text { and }(-2 \sqrt{2}, 4 \sqrt{6}) $$

Multiply, and then simplify each product. Assume that all variables represent positive real numbers. See Example 1. $$ (\sqrt{p}+5 \sqrt{s})(\sqrt{p}-5 \sqrt{s}) $$
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