Understanding Rational Exponents and Roots

The video tutorial explains how to solve equations with fractional exponents. It covers two examples: solving x^(2/3) = 64 and x^(4/3) = 9. The tutorial demonstrates two methods for solving these equations: using rational exponents and using roots. It emphasizes the importance of considering both positive and negative solutions when dealing with even exponents. The video also provides step-by-step instructions for simplifying expressions and using a calculator to verify results.

What is the first step in solving the equation x^(2/3) = 64?

• Take the square root of both sides

• Divide both sides by 2

• Square both sides

• Cube both sides

When solving x^2 = 64^3, why do we consider both positive and negative solutions?

• Because the exponent is even

• Because x is a variable

• Because 64 is a perfect square

• Because 64 is a large number

How many solutions are there for x when solving x^2 = 64^3?

• Two solutions

• Three solutions

• One solution

• No solution

What is the result of 64 raised to the power of 3/2?

• 1024

• 512

• 128

• 256

Why is it important to include a plus or minus sign when solving equations with even exponents?

• To account for both positive and negative roots

• To simplify the equation

• To make the equation more complex

• To ensure the solution is a whole number

In the equation x^(4/3) = 9, what is the first step to eliminate the fractional exponent?

• Take the fourth root of both sides

• Square both sides

• Divide both sides by 3

• Cube both sides

What is the reciprocal of 4 when solving x^4 = 9^3 using rational exponents?

• 1/5

• 1/4

• 1/3

• 1/2

What is the simplified form of the fourth root of nine cubed?

• 3 times the fourth root of 9

• 9 times the fourth root of 3

• 3 times the square root of 9

• 9 times the square root of 3

What operation is used to solve x^4 = 9^3 using radicals?

• Fifth root

• Cube root

• Square root

• Fourth root

What is the final solution in radical form for x^4 = 9^3?

• x = ±3 times the square root of 9

• x = ±9 times the fourth root of 3

• x = ±3 times the fourth root of 9

• x = ±9 times the square root of 3