Index notation and index laws 1 and 2

How many copies of x are multiplied here?

Count each occurrence of x as one factor

Evaluate this power. The index counts factors, not an ordinary multiplier

Write the base as three factors before multiplying

The brackets make negative 3 the base. Evaluate this power

Now use three factors of negative 3. What is the value?

Multiply the first two negative factors, then multiply by the third

Here the minus sign is outside the power. What is the value?

Predict the index when these two groups of x factors are joined

Count the factors in both powers together

Each tile represents one x. Multiply the factors within each group. Join the first two groups to form the product. Start with first index two and second index three, then try first index one and second index four, and first index four and second index two. Compare the joined count with adding and multiplying the indices

Which rule counts the joined factors for every pair?

Products of matching bases

For positive integer indices, joining repeated factors adds their counts. Keep the base unchanged. Multiply numerical coefficients separately

1. 1Same base Combine indices only for matching bases \(\displaystyle x^{2}\times x^{3}=x^{5}\)
2. 2Separate coefficient and index Multiply 5 by 2, then add 2 and 3 \(\displaystyle 5a^{2}\times2a^{3}=10a^{5}\)
3. 3Different bases These factors stay separate; adding their indices is not valid \(\displaystyle x^{2}y^{3}\)

Find only the numerical coefficient of this product

Ignore the letters for this step and multiply the two numbers

Combine the coefficient and index

The bases are different. Which expression preserves the product?

For x nonzero, cancel three matching factors. What index remains?

Cancel the three denominator factors from the seven numerator factors

Which input is excluded from this original quotient?

Quotients retain their domain

For a nonzero base, cancellation subtracts the denominator index from the numerator index. In this section the remaining index is positive; zero and negative indices follow in the next sections

1. 1Subtract in the correct order Take the denominator count from the numerator count \(\displaystyle 7-3=4\)
2. 2Divide coefficients A fractional coefficient is a valid exact result \(\displaystyle 3\div4=\frac34\)
3. 3Keep the restriction The simpler formula does not restore an excluded input \(\displaystyle \frac{x^{7}}{x^{3}}=x^{4},\quad x\ne0\)

For m nonzero, find only the numerical coefficient of the simplified quotient

Divide the numerator coefficient by the denominator coefficient

For m nonzero, select the complete simplified quotient

For m nonzero, add the numerator indices first, then subtract the denominator index. What index remains?

Add the two numerator indices first, then subtract the denominator index

For a nonzero, combine the denominator product before subtracting. What index remains?

Add both denominator indices before subtracting their total

For x nonzero, simplify only the coefficient in this expression

Reduce the ratio of the numerical coefficients

For x nonzero, what is the resulting index of x?

Add the numerator indices and denominator indices separately, then subtract

For x nonzero, assemble the simplified expression

Independent transfer: for p nonzero, simplify this new quotient