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Chapter 5: Problem 15

Find the value of \(t\) in each equation. \(3^{t}=729\)

Short Answer

Expert verified
The value of t is 6.

Step by step solution

01

Recognize the equation type

Identify that the equation is exponential with the variable in the exponent: \[ 3^{t} = 729 \]
02

Express 729 as a power of 3

Rewrite 729 as a power of the base 3. \ 729 = 3^{6} \ Then, we have: \[ 3^{t} = 3^{6} \]
03

Equate the exponents

Since the bases are the same, equate the exponents: \[ t = 6 \]

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Key Concepts

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

Exponentiation
When dealing with exponential equations, it is important to understand exponentiation, which is the operation of raising one number (the base) to the power of another number (the exponent). In the given equation, \(3^{t} = 729\), the base is 3 and the exponent is variable 't'. Exponentiation allows us to express large numbers compactly. For example, \(3^{6}\) means multiplying 3 by itself six times. Using this operation efficiently helps in solving exponential equations.
Equating Exponents
Once you have an equation with the same base on both sides, you can equate the exponents. This is because, by definition, if \(a^{x} = a^{y}\), then \(x = y \). In the exercise, after recognizing that 729 can be written as \(3^{6}\), the equation becomes \(3^{t} = 3^{6}\). Now, we can equate the exponents, giving \(t = 6\). This step simplifies our problem and allows us to solve for the unknown exponent.
Algebraic Manipulation
Algebraic manipulation involves rearranging equations to isolate the variable we want to solve for. In the case of exponential equations, once the exponents are equated, the solution involves simple algebraic steps. Here, solving the equation \(3^{t} = 3^{6}\) by equating exponents and isolating 't' gives us the final solution \(t = 6\). Mastery of algebraic manipulation is crucial for solving various mathematical problems efficiently. Remember to always check your work to ensure the solution is correct.

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Most popular questions from this chapter

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