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Chapter 2: Problem 136

Show that \((23-8 \sqrt{7})^{1 / 2}=4-\sqrt{7}\)

Short Answer

Expert verified
We showed that \((4-\sqrt{7})^2 = 23 - 8\sqrt{7}\) by applying the square of a binomial formula and simplifying. Thus, \((23-8 \sqrt{7})^{1 / 2}=4-\sqrt{7}\).

Step by step solution

01

Square \((4 - \sqrt{7})^2\)

Calculate the square of the expression \(4-\sqrt{7}\) using the square of a binomial formula: \((a-b)^2 = a^2 - 2ab + b^2\). In our case, \(a = 4\) and \(b = \sqrt{7}\).
02

Apply the formula

According to the formula, we have \((4 - \sqrt{7})^2 = 4^2 - 2 (\cdot) 4 (\cdot) \sqrt{7} + (\sqrt{7})^2\)
03

Calculate the terms

Now, calculate each individual term: \(4^2 = 16\), \(-2 (\cdot) 4 (\cdot) \sqrt{7} = -8 \sqrt{7}\), \((\sqrt{7})^2 = 7\)
04

Combine the terms

Combine the results from Step 3: \((4 - \sqrt{7})^2 = 16 - 8 \sqrt{7} + 7\)
05

Simplify the combined expression

Simplify the expression from Step 4: \((4 - \sqrt{7})^2 = 23 - 8 \sqrt{7}\)
06

Conclusion

As we have shown that \((4-\sqrt{7})^2 = 23-8\sqrt{7}\), we can conclude that the square root of the original expression \((23-8 \sqrt{7})^{1 / 2}\) is equal to \(4-\sqrt{7}\).

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Key Concepts

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

Square of a Binomial Formula
Understanding how to expand the square of a binomial is fundamental in algebra. A binomial is an algebraic expression containing two terms, such as \(a + b\) or \(a - b\). When squaring a binomial, the resulting expression has three parts: the square of the first term, twice the product of both terms, and the square of the second term. The square of a binomial formula is expressed as \( (a \pm b)^2 = a^2 \pm 2ab + b^2 \).

Binomial Expressions
Binomial expressions are expressions that consist of two terms separated by a plus or minus sign, such as \(a + b\) or \(a - b\). These are important in algebra, as they appear frequently in various mathematical contexts. Simplifying a binomial involves combining like terms and performing arithmetic operations as per the rules of algebra. Understanding how to manipulate these expressions, including addition, subtraction, and in particular squaring, is crucial for solving a wide range of problems.

Square Roots
Square roots are a fundamental concept in mathematics, represented by the radical symbol \( \sqrt{ } \). The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3, as \(3 \times 3 = 9\). Simplifying square roots involves finding the prime factors of the number and reducing it to the simplest radical form, if possible. In the context of binomials, knowing how to properly combine and simplify square roots is crucial when dealing with square root expressions, such as \(\sqrt{a^2} = a\) where \a\ is a nonnegative real number.

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Most popular questions from this chapter

Evaluate \(\left(\frac{1}{9}\right)^{x}\) if \(x\) is a number such that \(3^{x}=5\)

Find a formula for the inverse function \(f^{-1}\) of the indicated function \(f\). $$ f(x)=32 x^{5} $$

Evaluate the indicated quantities. Do not use a calculator-pushing buttons for these exercises will not help you understand rational powers. $$ (-27)^{4 / 3} $$

What is the domain of the function \(\left(1+x^{2}\right)^{1 / 8} ?\)

Find all real numbers \(x\) that satisfy the indicated equation. $$ x^{2 / 3}+3 x^{1 / 3}=10 $$
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