Index laws 3 to 5 and the zero power

Expand the brackets mentally. How many factors of x are there?

Write three copies of x squared and count the factors

With x equal to 2, compare the three rows. Try n equal to 1, then 3, then 0. The first two rows raise a whole bracket to n; the last raises only x

Which index rule fits the first two rows for every setting?

A power of a power

Repeat the entire inner power. Multiply the two indices because each repeated group has the same number of factors

1. 1Every factor inside the brackets A product power also raises its numerical coefficient. These rules here use positive whole-number indices \(\displaystyle (ab)^{n}=a^{n}b^{n}\)

Find the new index on p

Multiply the inner and outer indices

Square the coefficient before combining the factors

Treat the 4 as a factor that occurs twice

For the same expression, find the new index on p

Count the p factors in the two repeated groups

Combine the coefficient and the power of p

Apply the outer index to all three factors

Does a power distribute over addition? Evaluate this expression to test that idea

Power of a quotient

Raise both the numerator and denominator. A denominator must be nonzero; keep that restriction after simplifying

1. 1Original denominator In this rule, b cannot be zero. The numerator may be zero when n is positive \(\displaystyle b\ne0\)

Square the numerical fraction. Assume b is nonzero. What is the new coefficient?

Raise the numerator and denominator coefficients to the outer power

What is the new index on b in the denominator? Assume b is nonzero

The denominator b is repeated twice

Expression

\(\displaystyle \left(\frac{3a^{2}}{2b}\right)^{2}\)

Restriction

\(\displaystyle b\ne0\)

Now combine the powered numerator and denominator

Which restriction is required for the original quotient?

Divide the same nonzero quantity by itself. What is the value?

Evaluate the numerator and denominator first

Why a zero power is one

For a nonzero base, dividing a power by itself gives 1. Subtracting its equal indices gives zero. So a zero power must have value 1

1. 1The condition matters The base must be nonzero. The expression zero to the power zero is not defined by this rule \(\displaystyle a\ne0\)
2. 2Only its own base The 5 is outside the power and stays as a coefficient \(\displaystyle 5x^{0}=5\quad(x\ne0)\)

The brackets make negative 3 the whole base. Evaluate

Apply the zero-power rule to the whole bracketed base

Only x has index zero. Assume x is nonzero

Identify exactly which base carries the zero index

Now the entire product carries the zero power. Assume x is nonzero

Check whether the zero index applies to x alone or to the bracketed product

Both bases are nonzero. Evaluate the sum

Evaluate each power separately, then add

For which value of x can we use the ordinary zero-power rule?

The product is squared and then squared again. Find the coefficient. Assume z is nonzero

Follow both nested powers; first square 2, then square the result

For the same nested product, find the final index on x. Assume z is nonzero

Track the original index and both repeated groups

Expression

\(\displaystyle \left((2x^{2}y)^{2}\right)^{2}z^{0}\)

Restriction

\(\displaystyle z\ne0\)

Recombine all factors of the nested product

Expression

\(\displaystyle \left(\frac{(a^{2})^{3}}{b^{2}}\right)^{2}q^{0}\)

Restriction

\(\displaystyle b\ne0,\quad q\ne0\)

Only the numerator has an extra inner power. Simplify

Expression

\(\displaystyle \left(\frac{2r^{3}}{3s^{2}}\right)^{2}t^{0}\)

Restriction

\(\displaystyle s\ne0,\quad t\ne0\)

Independent check: simplify, retaining the given restrictions