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Chapter 9: Problem 58

Solve each equation or inequality. Check your solutions. $$ 3^{5 n+3}=3^{33} $$

Short Answer

Expert verified
The solution is \(n = 6\).

Step by step solution

01

Understand the Given Equation

We are given the equation \(3^{5n+3} = 3^{33}\). Since the bases on both sides of the equation are the same and equal to 3, we can equate the exponents.
02

Equate the Exponents

Since the bases are the same, set the exponents equal to each other: \(5n + 3 = 33\).
03

Solve for n

Subtract 3 from both sides of the equation: \(5n = 30\). Now, divide both sides by 5 to get \(n = 6\).
04

Verify the Solution

Substitute \(n = 6\) back into the original equation to check the solution: \(3^{5(6)+3} = 3^{30+3} = 3^{33}\). Since both sides are equal, the solution is verified.

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Key Concepts

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

Exponents
Exponents make working with repeated multiplication easier. When you see an expression with an exponent, like \(a^b\), it means you multiply the base \(a\) by itself \(b\) times. For example, \(3^4\) means \(3 \times 3 \times 3 \times 3\). Exponents are powerful tools in mathematics. They allow for expressing very large numbers in a compact form.
Exponents follow specific rules that make them easier to work with.:
  • Multiplication of the same base: When multiplying numbers with the same base, you can add the exponents: \(a^b \times a^c = a^{b+c}\).
  • Division of the same base: When dividing numbers with the same base, you subtract the exponents: \(a^b \div a^c = a^{b-c}\).
  • Power of a power: When raising an exponent to another power, you multiply the exponents: \((a^b)^c = a^{b \cdot c}\).
These rules are essential for simplifying expressions and solving equations involving exponents.
Equal Bases
When working with exponent equations, having equal bases can simplify things significantly. If you have an equation where both sides have the same base, like in \(3^{5n+3} = 3^{33}\), you can set the exponents equal to each other. This is because the only way for two powers with the same base to be equal is if their exponents are also equal.
Here鈥檚 why equal bases are helpful:
  • It reduces the complexity of the problem since you only need to focus on the exponents.
  • It transforms an otherwise potentially complicated exponential problem into a much simpler algebraic equation.
  • This method is particularly useful when solving exponential equations, as it often allows for bypassing some more intricate methods of solving complexity.
Whenever you see equal bases, it鈥檚 a good hint that you can simplify your work considerably by focusing on the exponents.
Solving Equations
Solving equations with exponents involves isolating the unknown variable. The process typically starts by setting up the equation with the exponents on one side and the constant terms on the other. For example, in the equation \(5n + 3 = 33\), isolating \(n\) requires a few simple algebraic steps:
  • Subtract the constant: Remove any constant on the variable's side by using subtraction, like \(5n = 33 - 3\).
  • Divide by the coefficient: Finally, isolate the variable by dividing both sides by the coefficient, which is \(5\) in this case, giving \(n = 6\).
This method is straightforward and follows basic principles of algebra. Keep performing these inverse operations鈥攁dding becomes subtracting, and multiplying becomes dividing鈥攗ntil the variable stands alone on one side of the equation.
Verifying Solutions
After solving an equation, it's crucial to verify your results to ensure accuracy. Verification involves plugging the solution back into the original equation to see if both sides are equal.
Take our solution of \(n = 6\) from the equation \(3^{5n+3} = 3^{33}\):
  • Substitute \(n = 6\) back into the left-hand side of the equation: \(3^{5(6)+3}\).
  • Simplify to \(3^{30+3} = 3^{33}\).
  • Check that the expression equals the right-hand side, \(3^{33}\).
Since both sides are equal, the value \(n=6\) is confirmed as the correct solution. Verifying solutions ensures that you haven't made any mistakes in your calculations, providing confidence in your answer. Remember, verification is a quick process that reinforces the reliability of your solutions.

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Most popular questions from this chapter

For Exercises \(33-35,\) use the following information. A small corporation decides that 8\(\%\) of its profits would be divided among its six managers. There are two sales managers and four nonsales managers. Fifty percent would be split equally among all six managers. The other 50\(\%\) would be split among the four nonsales managers. Let \(p\) represent the profits. Write an expression to represent the share of the profits each nonsales manager will receive.

State whether each equation represents a direct, joint, or inverse variation. Then name the constant of variation. \(y=-7 x\)

For Exercises 53 and \(54,\) use the following information. The musical cent is a unit in a logarithmic scale of relative pitch or intervals. One octave is equal to 1200 cents. The formula to determine the difference in cents between two notes with frequencies \(a\) and \(b\) is \(n=1200\left(\log _{2} \frac{a}{b}\right)\). Find the interval in cents when the frequency changes from 443 Hertz \((\mathrm{Hz})\) to 415 \(\mathrm{Hz}\) .

For Exercises 55 and 56 , use the following information. If you deposit \(P\) dollars into a bank account paying an annual interest rate \(r\) (expressed as a decimal), with \(n\) interest payments each year, the amount \(A\) you would have after \(t\) years is \(A=P\left(1+\frac{r}{n}\right)^{n t} .\) Marta places \(\$ 100\) in a savings account earning 2\(\%\) annual interest, compounded quarterly. If Marta adds no more money to the account, how long will it take the money in the account to reach \(\$ 125 ?\)

Suppose you deposit A dollars in an account paying an interest rate of r, compounded continuously. Write an equation giving the time t needed for your money to double, or the doubling time.
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