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Chapter 13: Problem 34

Solve each exponential equation. $$4^{3 a}=64$$

Short Answer

Expert verified
The short answer for the exponential equation \(4^{3 a} = 64\) is \(a = 1\).

Step by step solution

01

Rewrite the right-hand side with the same base as the left side of the equation

We are given the equation: \(4^{3 a} = 64\). Our goal is to express \(64\) as an exponent of \(4\). We know that \(4^3 = 64\). So we can rewrite the equation as: \(4^{3 a} = 4^3\)
02

Compare the exponents

Since the bases on both sides of the equation are the same (both are \(4\)), we can set the exponents equal to each other: \(3 a = 3\)
03

Solve for the variable \(a\)

Now we solve for the variable \(a\): \(3 a = 3\) Divide both sides by \(3\): \(a = 1\)
04

Final Answer

We have found the value of \(a\): \(a = 1\).

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Key Concepts

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

Solving Exponential Equations
When solving exponential equations, the key is to express both sides of the equation with the same base. This simplifies comparisons of the exponents. An exponential equation consists of variables in the exponent form, such as the given equation, \(4^{3a} = 64\).
  • The first step is to express both sides with identical bases. In our equation, we transform 64 into \(4^3\), as they both equal 64 when calculated.
  • Once you have the same base on each side, you equate the exponents. This is because if the bases are the same, the only way for the equality to hold is if the exponents are also equal.
  • Finally, you solve for the variable in the exponent using basic algebraic methods like dividing or multiplying both sides of the equation.
Exponential equations often need simplifying before they can be solved. Understanding the relationships between powers and exponents is crucial.
Exponents
Exponents represent the number of times a base is multiplied by itself. In our equation \(4^3 = 64\), 4 is the base and 3 is the exponent. This means \(4\) is multiplied by itself three times: \(4 \times 4 \times 4\).
Some essential rules for working with exponents are:
  • **Same Base Rule:** If \(a^m = a^n\), then \(m = n\).
  • **Power of a Power:** \((a^m)^n = a^{m\cdot n}\).
  • **Product of Powers:** \(a^m \cdot a^n = a^{m+n}\).
  • **Quotient of Powers:** \(\frac{a^m}{a^n} = a^{m-n}\).
Understanding these rules simplifies calculations and allows easier manipulation of exponential equations. Similarly, being comfortable with powers such as squares \(n^2\) and cubes \(n^3\) helps in efficiently rewriting expressions during problem-solving.
Algebra Basics
Algebra is all about finding the unknowns represented by variables in equations. In our example \(3a = 3\), it's clear and simple to solve.
Here's how we approach it:
  • You can isolate the variable by performing the same operation on both sides of the equation. In \(3a = 3\), divide both sides by 3 to isolate \(a\).
  • This operation yields \(a = 1\), fulfilling the equation's requirement.
  • These steps involve applying basic algebraic principles, like the distributive, associative, or commutative properties as needed in more complex problems.
Mastering these fundamental techniques ensures you can solve a variety of equations confidently. Always perform error checks by plugging your solution back into the original equation to verify that they satisfy the problem conditions.

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Most popular questions from this chapter

Find the inverse of each one-to-one function. Then, graph the function and its inverse on the same axes. $$h(x)=x+3$$

Determine whether each statement is true or false. If it is false, rewrite the statement so that it is true. Let \(f(x)\) be one-to-one. If \(f(7)=2,\) then \(f^{-1}(2)=7\)

Find the inverse of each one-to-one function. $$h(x)=\sqrt[3]{x-7}$$

The hydronium ion concentrations, \(\left[\mathrm{H}^{+}\right],\) are given for some common substances. Find the \(\mathrm{pH}\) of each substance (to the tenths place), and determine whether each substance is acidic or basic. $$\text { Egg white: }\left[\mathrm{H}^{+}\right]=2 \times 10^{-8}$$

Write as a single logarithm. Assume the variables are defined so that the variable expressions are positive and so that the bases are positive real numbers not equal to \(1 .\) $$3 \log a+4 \log c-6 \log b$$
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