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Chapter 3: Problem 49

Suppose \(x\) is a number such that \(2^{x}=5 .\) Evaluate \(8^{x}\).

Short Answer

Expert verified
Thus, we find that \(8^{x} = 125\).

Step by step solution

01

Rewrite \(8^{x}\) in terms of \(2^{x}\)

We know \(8 = 2^3\), so we can rewrite \(8^{x}\) as \((2^3)^{x}\). Now, using the power of a power property, we can simplify this expression to \(2^{3x}\).
02

Substitute the given relationship

Now that we have \(8^{x}\) rewritten as \(2^{3x}\), we can substitute the given relationship \(2^{x}=5\) into the expression. We do this by replacing \(2^{x}\) with \(5\), so we have \((5)^{3}\).
03

Evaluate the expression

Now we just need to evaluate \((5)^{3}\), which means \(5 \times 5 \times 5 = 125\). So, \(8^{x} = \boxed{125}\).

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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 crucial. This rule states that when you raise a power to another power, you multiply the exponents. For example, if you have \( (a^m)^n \), it simplifies to \( a^{mn} \). This makes calculations more efficient and helps you simplify expressions quickly.

In our example, we used this property while rewriting \( 8^x = (2^3)^x \). Applying the property gave us \( 2^{3x} \). Breaking down complex powers into simpler ones can make solving exponential equations much easier.
  • This property helps in reducing multiple steps to one simple calculation.
  • It is useful in both algebra and higher-level mathematics.
  • Recognizing when to apply this rule helps solve exponential equations efficiently.
Exponentiation
Exponentiation is a mathematical operation involving two numbers: the base and the exponent. The base is the number that is multiplied by itself, and the exponent tells how many times this multiplication occurs. For example, \( a^n \) means multiplying \( a \) by itself \( n \) times.

In the problem we've solved, understanding the meaning of \( 2^x = 5 \) and evaluating \( 8^x \) by rewriting it as \( 2^{3x} \) was key. We then multiplied \( 5 imes 5 \times 5 \) to get 125, showing how exponentiation transforms multiplication into a more concise operation.
  • Exponentiation is frequently used in growth calculations, such as compound interest.
  • It simplifies repeated multiplication.
  • Understanding exponentiation lays the foundation for more complex concepts, like logarithms.
Substitution Method
The substitution method is a powerful tool in algebra to simplify problems by replacing one part of an equation with a known value. In the exercise, we substituted \( 2^x = 5 \) into \( 2^{3x} \) to make calculations straightforward.

This approach can transform a seemingly complex task into a simple arithmetic problem.
It's particularly effective in solving systems of equations or evaluating expressions based on given conditions.
  • By substituting known values, we reduce variables and simplify expressions.
  • This method is not only used in solving equations but also in calculus for integrating functions.
  • Practicing substitution helps improve problem-solving skills across various mathematical disciplines.

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Most popular questions from this chapter

Suppose \(x\) is a positive number. Using only the definitions of roots and integer powers, explain why $$ \left(x^{1 / 2}\right)^{3}=\left(x^{1 / 4}\right)^{6}. $$

8uppose \(\$ 8000\) is deposited in a bank account paying \(7 \%\) interest per year, compounded 12 times per year. How much will be in the bank account at the end of 100 years?

Write a description of the logarithmic scale used for the \(\mathrm{pH}\) scale, which measures acidity (this will probably require use of the library or the web).

Suppose you have a calculator that can only compute square roots. Explain how you could use this calculator to compute \(7^{1 / 8}\).

Suppose a television is playing softly at a sound level of 50 decibels. What decibel level would make the television sound eight times as loud?
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