•Explain how writing 2x-5 with positive exponents is similar to writing 2x-3/4 with positive exponents. Post a problem involving a rational exponent for your classmates to solve.
2x^-5 = 2/x^5
2x^(-3/4) = 2/x^(3/4)
a negative exponent just means shifting the value between numerator and denominator. Note that this also means that
2/x^(-2) = 2x^2
since that is 2 / (1/x^2) = 2 * x^2/1 = 2x^2
In both cases, when we convert the expressions with negative exponents to positive exponents, we need to follow the rule that states:
If we have an expression with a negative exponent, we can rewrite it by moving the term with the negative exponent to the denominator and changing the sign of the exponent to positive.
So, let's look at the examples you provided:
1. Writing 2x-5 with positive exponents:
We have 2x raised to the power of -5. To convert it to positive exponents, we move the term with the negative exponent to the denominator:
1 / (2x)^5
2. Writing 2x-3/4 with positive exponents:
We have 2x raised to the power of -3/4. Following the rule, we move the term with the negative exponent, 2x, to the denominator and change the sign of the exponent to positive:
1 / (2x)^(3/4)
Both examples demonstrate how to convert expressions with negative exponents to positive exponents by moving the terms with negative exponents to the denominators and changing the signs of the exponents.
Now, as for a problem involving a rational exponent for your classmates to solve, here's an example:
Solve for x: (16x^(2/3))^3 = 64
To solve this problem, we can follow these steps:
1. Cube the base, 16x^(2/3):
(16x^(2/3))^3 = 16^3 * (x^(2/3))^3 = 4096x^2
2. Set the resulting expression equal to the right side of the equation:
4096x^2 = 64
3. Solve the equation for x:
Divide both sides by 4096: x^2 = 64/4096 = 1/64
4. Take the square root of both sides to isolate x:
x = ± √(1/64) = ± 1/8
So, the solutions to the equation are x = 1/8 and x = -1/8.