Problem 103 Simplify. Assume that \(x \geq 0... [FREE SOLUTION]

Chapter 10: Problem 103

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

Short Answer

Expert verified

\ 4 \sqrt{3} \

Step by step solution

- Rewrite the given expression

Rewrite \ \( \sqrt[4]{48^{2}} \) as \ \( \left( 48^{2} \right)^{\frac{1}{4}} \ \). This utilizes the property of radicals and exponents.

- Apply the power rule

Using the power rule \ \( (a^{m})^{n} = a^{mn} \ \) gives us \ \( 48^{2 \cdot \frac{1}{4}} = 48^{\frac{2}{4}} = 48^{\frac{1}{2}} \ \).

- Simplify the exponent

Simplify the exponent \ \( 48^{\frac{1}{2}} \ \) which is the same as \ \( \sqrt{48} \ \).

- Simplify the square root

Factor \ \( 48 \ \) into its prime factors to find \ \( 48 = 16 \cdot 3 = (4^2) \cdot 3 \ \). Thus, \ \( \sqrt{48} = \sqrt{4^2 \cdot 3} = 4 \sqrt{3} \ \).

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Key Concepts

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

Exponents

Exponents are a way to express repeated multiplication of a number by itself. For example, when you see \(2^3\), it means 2 is multiplied by itself three times, so it equals 2 * 2 * 2 which is 8.
Using exponents makes calculations more efficient. Instead of writing a long multiplication, you simply write the base number and the power.
One important rule is the power rule: \((a^m)^n = a^{mn}\). This means if you have an exponent raised to another exponent, you multiply the exponents together.
In the exercise, we used this rule when simplifying \((48^2)^{1/4} \). We multiplied 2 by 1/4 to get 1/2.

Radicals

Radicals, also known as roots, are the inverse operation of exponents. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 * 4 = 16.
In mathematical notation, the square root of a number x is written as \(\backslashsqrt{x}\). There are also higher-order roots, like the cube root \( \backslashsqrt[3]{x} \) and fourth root \( \backslashsqrt[4]{x} \).
In the exercise, we dealt with the fourth root of 48 squared, written as \(\backslashsqrt[4]{48^2} \). By converting this to a fractional exponent \((48^{1/2}) \), we can simplify the number more easily.

Prime Factorization

Prime factorization breaks a number down into its basic building blocks, which are prime numbers. Prime numbers are numbers greater than one that have no divisors other than 1 and itself, like 2, 3, 5, 7, and so on.
To find the prime factors of a number, you divide the number by prime numbers until you can鈥檛 divide anymore. For example, the prime factorization of 48 is 2 * 2 * 2 * 2 * 3, or more compactly, \( 2^4 * 3 \).
This method was used in our exercise to simplify \(\backslashsqrt{48} \) as \(\backslashsqrt{4^2 * 3} \) and helped us further simplify it to \ 4\backslashsqrt{3} \.

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91影视

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

Short Answer

Step by step solution

- Rewrite the given expression

- Apply the power rule

- Simplify the exponent

- Simplify the square root

Key Concepts

Exponents

Radicals

Prime Factorization

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