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Chapter 10: Problem 33

Evaluate or simplify each expression. (Assume \(x \geq 0 .)\) $$(\sqrt{7})^{4}$$

Short Answer

Expert verified
The expression \( (\sqrt{7})^{4} \) simplifies to 49.

Step by step solution

01

- Understand the Problem

The expression \((\sqrt{7})^{4}\) involves raising a square root to the fourth power. Simplifying this requires applying the rules of exponents and radicals.
02

- Use Exponent Rules

Recall that \((a^{m})^{n} = a^{m \cdot n}\). Here, \(a\) is \(\sqrt{7}\), \(m\) is 1/2 (since \(\sqrt{7}=7^{1/2}\)), and \(n\) is 4. This gives us: \((\sqrt{7})^{4} = (7^{1/2})^{4}\).
03

- Simplify the Exponent

Apply the exponent rule to get: \((7^{1/2})^{4} = 7^{1/2 \cdot 4} = 7^{2}\).
04

- Calculate the Power

Next, compute \(7^{2}\). This means multiplying 7 by itself: \(7^{2} = 7 \cdot 7 = 49\).
05

- Write the Final Answer

Thus, the expression \( (\sqrt{7})^{4} \) simplifies to 49.

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Key Concepts

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

Exponent Rules
Exponent rules help simplify expressions involving powers. One key rule is \((a^m)^n = a^{mn}\). This tells us that when raising a power to another power, we multiply the exponents together.
This rule is particularly useful when dealing with roots and fractional exponents.
For example, if we start with \(\sqrt{7}\), it's the same as \(7^{1/2}\).
Raising \(\sqrt{7}\) to the 4th power translates to \( (7^{1/2})^4 \). According to our rule, you multiply the exponents:
\[ (7^{1/2})^4 = 7^{(1/2)*4} = 7^2\]
Understanding and applying exponent rules can simplify complex expressions efficiently.
Radicals
Radicals, often denoted with the square root symbol (\(\sqrt{}\)), represent roots of numbers. For example, \( \sqrt{7} \) is the square root of 7.
A vital connection between radicals and exponents is that any root can be expressed as a fractional exponent. The square root of a number \(n\) is \(n^{1/2}\).
This allows us to use exponent rules to simplify expressions involving roots.
For the expression \( \sqrt{7} \), raising it to a power is easier if we express it in exponential form first:
\(\sqrt{7} = 7^{1/2} \).
This makes it straightforward to apply exponent rules for further simplification.
Powers of Numbers
Powers of numbers indicate repeated multiplication. For example, \(7^2 = 7 \cdot 7\).
When solving problems involving powers, it's key to remember what these expressions mean.
For instance, the solution \( (\sqrt{7})^{4}\) simplifies to \( 7^2\), because \(\sqrt{7} = 7^{1/2} \).
Once rewritten, applying the power of 4 means multiplying the exponents together:
\[ (7^{1/2})^4 = 7^{(1/2) \cdot 4} = 7^2 = 49 \].
The final answer shows how repeated multiplication works. Practicing these conversions enhances understanding of both powers and radicals.

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Most popular questions from this chapter

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$\sqrt{24}+\frac{5}{\sqrt{6}}$$

Solve the following problems algebraically. Be sure to label what the variable represents. Xavier made three investments at \(6.5 \%, 7.6 \%,\) and \(9.2 \% .\) The amount invested at \(7.6 \%\) is \(\$ 1000\) less than the amount invested at \(9.2 \%,\) and the amount invested at \(6.5 \%\) is twice the amount invested at \(7.6 \% .\) If the annual income from the three investments is \(\$ 837,\) how much is invested altogether?

Rationalize the denominators and simplify. $$(3+\sqrt{5})^{2}+\frac{8}{3-\sqrt{5}}$$

Rationalize the denominators and simplify. $$(4-\sqrt{7})^{2}-\frac{18}{4+\sqrt{7}}$$

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$\sqrt{20}-\sqrt{5}$$
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