Simplify using index laws 1 and 2

Two stages

\(\displaystyle 2^4\times2^3\)

A scaling process applies four factors of 2, then three more factors of 2. Which power gives the overall factor?

Each tile represents one factor x, not an amount to add. Change m and n. The product joins all factors from the first two groups

Which rule works for every pair of slider settings?

Count, cancel and keep the domain

For the same base, multiplication adds indices and division subtracts them. Cancel only nonzero factors. In a product or quotient with different letters, work on each letter separately

1. 1Add indices for the same base A product joins the factors from both powers. Keep the base unchanged \(\displaystyle a^3\times a^4=a^{3+4}=a^7\)
2. 2Subtract when dividing Cancel matching factors. The denominator must be nonzero \(\displaystyle \frac{a^8}{a^3}=a^{8-3}=a^5\quad(a\ne0)\)
3. 3Keep different bases separate Collect powers of each letter independently \(\displaystyle a^2b^5\times a^3b^2=a^5b^7\)
4. 4Calculate coefficients separately Multiply or divide numerical factors using ordinary arithmetic; a quotient coefficient can be a fraction \(\displaystyle 3x^2\times4x^3=12x^5\)
5. 5Complete cancellation gives one For a nonzero base, equal numerator and denominator powers cancel to 1. A resulting index 0 does not mean the whole expression is 0. \(\displaystyle \frac{a^3}{a^3}=a^0=1\quad(a\ne0)\)

Quotient

\(\displaystyle \frac{x^8}{x^3}\)

For x ≠ 0, simplify the quotient. Enter only the resulting index of x.

Assume a and b are nonzero. Select the first incorrect line

Let r ≠ 0. After combining and cancelling factors, what index of r remains? Enter the index, not the value of the whole expression

Simplification

\(\displaystyle \frac{x^5}{x^2}=x^3\quad(x\ne0)\)

Can the original quotient be evaluated at x = 0?