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Chapter 1: Problem 6

Simplify (without your GDC) each expression to a single integer. $$(\sqrt{2})^{6}$$

Short Answer

Expert verified
\((\sqrt{2})^6 = 8\).

Step by step solution

01

Understand the Expression

The expression given is \((\sqrt{2})^6\). This means that we are raising the square root of 2 to the 6th power.
02

Convert to Exponents

Recall that raising a square root to a power can be expressed using exponents. \(\sqrt{2}\) is equivalent to \(2^{1/2}\). So, \((\sqrt{2})^6\) becomes \((2^{1/2})^6\).
03

Apply the Power of a Power Property

Use the power of a power property \((a^{m})^{n} = a^{m \times n}\). Here \(m = 1/2\) and \(n = 6\). Applying this property gives us \(2^{1/2 \times 6}\).
04

Simplify the Exponent

Calculate the product of the exponents: \(1/2 \times 6 = 3\). Therefore, the expression simplifies to \(2^3\).
05

Calculate the Power

Evaluate \(2^3\) which means 2 multiplied by itself three times: \(2 \times 2 \times 2 = 8\).

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Key Concepts

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

Power of a Power Property
When dealing with exponents, the **Power of a Power Property** is a handy tool to know. This property states that when you raise an exponential expression to a power, you can multiply the exponents.

For example, if you have
  • \((a^m)^n\), it becomes \(a^{m \cdot n}\).
In our problem, we started with \((2^{1/2})^6\). According to the Power of a Power Property, we can multiply the exponents:
  • \(1/2 \times 6 = 3\).
This simplifies the expression to \(2^3\). This property makes complex exponential expressions much easier to handle by reducing them to a single exponent.
Simplifying Expressions
Simplifying expressions involves making them as simple as possible without changing their value. Often, this requires performing operations like multiplication, addition, or applying properties such as the Power of a Power Property.

For the expression \((\sqrt{2})^6\), simplifying means converting \(\sqrt{2}\) into exponential form first, and then applying the necessary rules to simplify further.

The steps were:
  • Convert \(\sqrt{2}\) to \(2^{1/2}\) by expressing the square root as an exponent.
  • Use the Power of a Power Property on \((2^{1/2})^6\), which simplifies to \(2^3\).
  • Finally, calculate \(2^3\) which equals 8.
Understanding each of these steps ensures clarity and correctness when handling exponential expressions.
Integer Powers
Calculating powers of integers is an essential skill in mathematics. An **integer power** means raising a whole number to another whole number.

For example, in our simplified expression \(2^3\), the base is 2, and the exponent is 3. This tells us to multiply 2 by itself three times:
  • \(2 \times 2 = 4\)
  • \(4 \times 2 = 8\)
Thus, \(2^3\) equals 8. Calculating integer powers often serves as a building block for more advanced math problems because it makes detailed calculations significantly easier. Understanding how to quickly compute these values is part of mastering both mathematical concepts and practical problem-solving.

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Most popular questions from this chapter

Use both inequality and interval notation to represent the given subset of real numbers. \(x\) is greater than or equal to 4 and less than 10

Rationalize the numerator, simplifying if possible. $$\frac{\sqrt{a}-3}{a-9}$$

Using properties of inequalities, prove each of the statements. a) If \(x0,\) then \(\frac{1}{y}<\frac{1}{x}\) b) If \(x<0\frac{1}{x}\)

Rationalize the denominator, simplifying if possible. $$\sqrt{\frac{1}{x^{2}}-1}$$

Determine whether each statement is true for all real numbers \(x\). If the statement is false, then indicate one counterexample, i.e. a value of \(x\) for which the statement is false. $$\frac{1}{x} \leqslant x$$
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