Exponential Equations and Properties

Questions and Answers

What is the value of x in the equation $2^x=16$?

What is the result of $(2^3)^x$?

2^{3x}

What expression simplifies to -1(3x-7) in the equation $(5^{-1})^{3x-7}$?

5^{-3x+7}

What does $(4^{-3})^{1-x^3}$ simplify to?

-3(1-x^2) = 4^{-3 + 3x^2} Signup and view all the answers

For the equation $2^{x-3}=64$, what is the value of x?

6 Signup and view all the answers

What is the final value of x in the equation derived from $2^{x-3}=64$?

9 Signup and view all the answers

Study Notes

Exponential Equations

Solve (2^x = 16) to find (x = 4) since (16) can be expressed as (2^4).

Exponent Properties

Using properties of exponents, ((2^3)^x) simplifies to (2^{3x}).

Negative Exponents

The expression ((5^{-1})^{3x - 7}) simplifies as follows:
- First, apply exponent rules to get (5^{-(3x - 7)}).
- This leads to the equation (-1(3x - 7) = 5^{-3x + 7}).

Manipulation of Negative Exponents

((4^{-3})^{1 - x^3}) can be manipulated:
- Transforms to (-3(1 - x^2) = 4^{-3 + 3x^2}).

Solving Variations of Exponential Equations

For (2^{x-3} = 64):
- Recognize that (64 = 2^6) to set the equation as (2^6 = 2^{x^2 - 3}).
- This gives (6 = x^2 - 3).
- By adding (3) to both sides, (9 = x^2), which leads to (x = 3) or (x = -3) (considering square roots).

Practice Critical Thinking

Review methods used for transforming and solving various exponential equations for a comprehensive understanding.

Understanding Exponential Rules

Ensure strong grasp of properties of exponents, including multiplication, division, and power rules, which are essential for simplifying expressions.

Description

This quiz covers various concepts related to exponential equations, including solving simple exponential equations and understanding the properties of exponents. You will also explore manipulations involving negative exponents and apply critical thinking to solve different variations of exponential equations. Test your knowledge and enhance your skills in handling exponential functions.