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# How do you simplify z^-2 / z^-3?

At this point it's confirmed that I suck at math.

### 21 Answers

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Hint

a^x * a^y = a^(x + y)

a^x / a^y = a^(x - y)

• Since the base is the same just subtract the exponents.

A negative exponent means you have a fraction

Go back to the method you used when dividing a common fraction by a common fraction

z^-2 = 1 / z^2

z^-3 = 1 / z ^3

Do you remember the rule: invert and multiply?

[1 / z^2] / [1 / z^3] = [1 / z^2] X [z^3 / 1]

• multiple z means snoozing.

• In a a ratio, if the bases are the same, subtract the exponents.

(-2) - (-3) = 1  therefore z to the first power. QED

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• First, invert both to get rid of negative exponents:

z⁻²/z⁻³ = z³/z²

Now reduce: (z²/z²)*z¹ = 1*z

= z

• Go back to law of exponents:

When multiplying two powers of the same base (here, the base is "z"), just add the powers.

example:  z^2 * z^4 = zz * zzzz = zzzzzz = z^6

z^2 * x^4 = z^(2+4) = z^6

When dividing two powers (of the same base), subtract the powers

z^5 / z^2 = zzzzz / zz = zz*zzz / zz = zzz = z^3

z^5 / z^2 = z^(5-2) = z^3

z^0 = 1

This comes from the second rule.

Anything divided by itself = 1

z^4 / z^4 = 1

z^4 / z^4 = z^(4-4) = z^0

A negative power is a "reciprocal" (a fraction)

1 / z^3 = z^0 / z^3 = z^(0-3) = z^(-3)

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Your problem  begins as a case for the second rule (division)

just be careful with signs

z^(-2) / z^(-3) = z^(-2 - -3) = z^(-2+3) = z^1 = z

• Remember z^-2 = 1/z^2

Similarly z^-3 = 1/z^3

Hence [ 1/z^2]/ [1/z^3]  is division of fractions.

Hence

[1/z^2] x [z^3/1]

Cancel down by 'z^2'

Hence

1/1 x z/1 = 'z^1' = 'z'   The answer

• Hardly anyone gets these right. Try multiplying numerator and denominator both by z³, but do be careful.

z⁻² / z⁻³

= (z⁻² / z⁻³)(z³ / z³) ... where z ≠ 0

= z³⁻² / z³⁻³

= z¹ / z⁰

= z

z⁻² / z⁻³ = z, where z ≠ 0

Without that exclusion, the answer z would be incorrect, or at least incomplete. In the first step, the denominator is multiplied by z³. If z were zero, that would be division by zero, an invalid operation. The given expression is undefined at z = 0. No amount of simplification can change that.

• Note that except for Daniel H answer, the others are wrong and made mistakes.

• When you have a negative exponent, you can swap it to the other side of the fraction and change the sign. So z^-2 would change to z^2 in the denominator. Likewise, z^-3 in the denominator would become z^3 in the numerator.

So this would turn into:

z^3 / z^2

At that point you can reduce this. When you divide two numbers (with the same base), you subtract the exponents:

= z^(3-2)

= z^1

= z

Answer:

z (as long as z ≠ 0)

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