Problem 138 Show that \((99+70 \sqrt{2})^{1 ... [FREE SOLUTION]

Chapter 2: Problem 138

Show that \((99+70 \sqrt{2})^{1 / 3}=3+2 \sqrt{2}\).

Short Answer

Expert verified

The short answer is: We showed that taking the cube root of the expression \((99+70\sqrt{2})\) is equal to \(3+2\sqrt{2}\) by first cubing the given result \((3+2\sqrt{2})^3\), then simplifying and combining the terms using the binomial formula. This led us to the expression \(99 + 70\sqrt{2}\), which matches the given expression, demonstrating that \((99+70\sqrt{2})^{1/3} = 3+2\sqrt{2}\).

Step by step solution

Cube the result.

First, we need to cube the given result \((3+2 \sqrt{2})^3\). Using the binomial formula \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\), where \(a = 3\) and \(b = 2 \sqrt{2}\), we get: \((3+2 \sqrt{2})^3 = 3^3 + 3(3^2)(2\sqrt{2}) + 3(3)(2\sqrt{2})^2 + (2\sqrt{2})^3\)

Simplify the expression.

Now, let's simplify the expression we got in Step 1: \(3^3 = 3 \times 3 \times 3 = 27\) \(3(3^2)(2\sqrt{2}) = 3 \times 9 \times 2\sqrt{2} = 54\sqrt{2}\) \(3(3)(2\sqrt{2})^2 = 3 \times 3 \times (2\sqrt{2} \times 2\sqrt{2}) = 9 \times 8 = 72\) \((2\sqrt{2})^3 = 2\sqrt{2} \times 2\sqrt{2} \times 2\sqrt{2} = 2\sqrt{2} \times 8 = 16\sqrt{2}\)

Combine the simplified terms.

Now that we have simplified the terms, we need to combine them: \((3+2\sqrt{2})^3 = 27 + 54\sqrt{2} + 72 + 16\sqrt{2} = 99 + 70\sqrt{2}\)

Compare the expressions.

In Step 3, we arrived at the expression \(99 + 70\sqrt{2}\). This is the same as the given expression \((99+70 \sqrt{2})^{1 / 3}\). Therefore, we have demonstrated that taking the cube root of \((99+70\sqrt{2})\) is equal to \(3+2\sqrt{2}\): \((99+70 \sqrt{2})^{1 / 3}=3+2 \sqrt{2}\)

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Key Concepts

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

Binomial Theorem

The Binomial Theorem is fundamental to understanding expressions raised to powers, particularly when dealing with radicals, like in our original exercise. It provides a way to expand expressions of the form \((a + b)^n\).
For a cube, specifically, it takes the form:

\(a^3\)
\(3a^2b\)
\(3ab^2\)
\(b^3\)

When we apply this to \((3 + 2 \sqrt{2})\), we let \(a\) be \(3\) and \(b\) be \(2 \sqrt{2}\). Each coefficient and term represents different parts or combinations of \(a\) and \(b\), allowing us to break down
complex expressions into manageable components.
By understanding this theorem, we can expand any binomial expression systematically, which is crucial for higher-level algebra.

Simplifying Expressions

Simplifying expressions involves rewriting them to their simplest form.
In mathematics, this is not just limited to removing parentheses or combining like terms. It鈥檚 about making the expression as understandable and direct as possible.
Take the example \((3 + 2 \sqrt{2})^3\). Using the binomial theorem, we ended up with terms: \(27\), \(54\sqrt{2}\), \(72\), and \(16\sqrt{2}\).
Simplifying these terms requires:

Calculating each power or multiplication step accurately.
Combining like terms, such as integers with integers and radicals with radicals.

This helps achieve a single, simplified expression that is easier to compare or further process. This practice is especially useful in solving equations and inequalities efficiently.

Radicals

Radicals, such as square roots, play a significant role in algebra,
especially when combined with other operations like addition or multiplication. Simplifying radicals can often involve breaking down the number or
expression within the root into smaller, simpler parts.
In the exercise, the term \(2 \sqrt{2}\) is treated as a unit, and its powers are calculated in steps:

First, calculate \((2 \sqrt{2})^2\).
Then, determine \((2 \sqrt{2})^3\).
Notice how each power multiplication stabilizes the radical part, resulting in whole numbers or simpler radical forms.

Mastering radicals involves knowing how to multiply and add them, transform them, and recognize their patterns.

Exponential Equations

Exponential equations involve expressions where variables appear in exponents, and solving them often requires using properties of exponents effectively. In our example, we indirectly dealt with exponential principles when discussing cube roots.
The cube root, \((x)^{1/3}\), is simply another way of expressing an exponent, with a fractional power indicating the root.

Understanding these roots and their corresponding exponents is key to manipulating equations across algebra.
Cube roots simplify exponential equations, transforming them into more tangible arithmetic expressions.

With all equations, practice identifying exponential, linear, or polynomial characteristics helps greatly in solving them efficiently.

91影视

Show that \((99+70 \sqrt{2})^{1 / 3}=3+2 \sqrt{2}\).

Short Answer

Step by step solution

Cube the result.

Simplify the expression.

Combine the simplified terms.

Compare the expressions.

Key Concepts

Binomial Theorem

Simplifying Expressions

Radicals

Exponential Equations

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