Zero, Negative, and Fractional Exponents

1) Deriving a^0 = 1 from the Quotient Rule

We want exponent rules to stay consistent even when the exponent is not a positive integer. Start with the quotient pattern for same-base powers:

a^m / a^n = a^{m-n} (for a \neq 0)

Now choose m = n. Then the left side becomes a number divided by itself:

• a^m / a^m = 1 (as long as a \neq 0)
• Using the exponent pattern: a^{m-m} = a^0

So consistency forces:

a^0 = 1 for every nonzero base a.

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Important notes

• 0^0 is not defined in typical algebra courses because different “consistent extensions” can conflict; avoid it unless a specific context defines it.
• a^0 is defined only for a \neq 0 in this rule-based derivation.

2) Negative Exponents as Reciprocals

Negative exponents are defined so the quotient pattern continues to work when the exponent difference is negative.

Example idea: if m < n, then m-n is negative. We still want a^m / a^n = a^{m-n} to be true.

Rewrite the fraction by “canceling” factors:

a^m / a^n = 1 / a^{n-m} (for a \neq 0)

But the exponent rule says this should equal a^{m-n}. Therefore we define:

a^{-(k)} = 1 / a^k for a \neq 0.

Practice: rewriting negative exponents to positive exponents

Rule of thumb: a negative exponent means “take the reciprocal.”

• a^{-3} = 1/a^3
• 1/a^{-3} = a^3
• (2x)^{-2} = 1/(2x)^2 = 1/(4x^2)

Step-by-step examples

Example 1: Rewrite 5x^{-4} with positive exponents.

• Move x^{-4} to the denominator as x^4: 5x^{-4} = 5/x^4

Example 2: Rewrite \frac{3}{y^{-2}} with positive exponents.

• y^{-2} in the denominator moves to the numerator as y^2
• \frac{3}{y^{-2}} = 3y^2

Example 3 (parentheses matter): Compare -2^{-3} and (-2)^{-3}.

• -2^{-3} = -(2^{-3}) = -(1/2^3) = -1/8
• (-2)^{-3} = 1/(-2)^3 = 1/(-8) = -1/8

They match here, but they do not always match for other exponents (especially even exponents), so keep track of parentheses.

3) Fractional Exponents as Roots

Fractional (rational) exponents connect exponents to radicals (roots). The goal is consistency with the power-of-a-power pattern:

(a^m)^n = a^{mn}

Defining a^{1/n}

We define a^{1/n} to mean “the number whose nth power is a.” In radical form:

a^{1/n} = \sqrt[n]{a}

Then:

(a^{1/n})^n = a

Defining a^{m/n}

Use the power-of-a-power idea:

a^{m/n} = (a^{1/n})^m = (\sqrt[n]{a})^m = \sqrt[n]{a^m}

Domain restrictions for real numbers

When working in real numbers, roots impose restrictions:

• If n is even (like 2, 4, 6), then \sqrt[n]{a} is real only when a \ge 0.
• If n is odd (like 3, 5), then \sqrt[n]{a} is real for all real a (including negative values).

This affects expressions like x^{1/2} (requires x \ge 0 for real outputs) versus x^{1/3} (allowed for all real x).

Negative fractional exponents

Combine the two ideas (negative exponent + fractional exponent):

a^{-m/n} = 1 / a^{m/n} = 1 / \sqrt[n]{a^m} (with the usual real-domain restrictions and a \neq 0 when the exponent is negative).

4) Simplifying Expressions with Radicals and Rational Exponents

Two equivalent languages are available: radicals and rational exponents. Choose the one that makes simplification clearer, but keep domain restrictions in mind.

A. Converting between forms

• \sqrt{x} = x^{1/2}
• \sqrt[3]{x^2} = x^{2/3}
• x^{5/2} = \sqrt{x^5} = (\sqrt{x})^5 = x^2\sqrt{x} (for real x \ge 0)

B. Simplifying step-by-step (examples)

Example 1: Simplify 16^{3/4}.

• Rewrite as a root: 16^{3/4} = (\sqrt[4]{16})^3
• \sqrt[4]{16} = 2 because 2^4 = 16
• So (\sqrt[4]{16})^3 = 2^3 = 8

Example 2: Simplify x^{7/3} (assume real x and note restrictions).

• x^{7/3} = x^{6/3} \cdot x^{1/3} = x^2 \cdot x^{1/3}
• Radical form: x^2\sqrt[3]{x}
• Since it is a cube root, this is real for all real x.

Example 3: Simplify \sqrt{50x^4} (real numbers).

• Factor perfect squares: 50x^4 = 25 \cdot 2 \cdot x^4
• \sqrt{25} = 5 and \sqrt{x^4} = |x^2| = x^2 (since x^2 \ge 0)
• So \sqrt{50x^4} = 5x^2\sqrt{2}

Example 4: Simplify \frac{x^{-1/2}}{x^{3/2}} (real x > 0).

• Combine exponents: x^{-1/2} / x^{3/2} = x^{-1/2 - 3/2} = x^{-4/2} = x^{-2}
• Rewrite with positive exponent: x^{-2} = 1/x^2
• Restriction: because of x^{1/2} in the original, we need x > 0 (also avoids division by zero).

C. Watch for parentheses and base signs

When the base is negative, rational exponents can be tricky in real numbers.

• (-8)^{1/3} = -2 is real because it is an odd root.
• (-8)^{1/2} is not real (square root of a negative).
• -8^{1/3} means -(8^{1/3}) = -2; parentheses decide whether the negative sign is part of the base.

5) Targeted Exercises (Mixed Types, Signs, and Parentheses)

Simplify each expression. Unless stated otherwise, work over the real numbers and state any necessary restrictions.

A. Zero and negative exponents

• 1) 7^0
• 2) (-3)^0
• 3) \frac{5x^3}{5x^3} (include restrictions)
• 4) 2x^{-5}
• 5) \frac{9a^{-2}}{3a}
• 6) (3y^2)^{-1}
• 7) \frac{1}{(2m)^{-3}}
• 8) \left(\frac{x}{y}\right)^{-2}

B. Fractional exponents and radicals

• 9) 27^{2/3}
• 10) 81^{3/4}
• 11) x^{5/2} (rewrite using radicals; give real-domain restriction)
• 12) \sqrt[3]{16x^5} (simplify by factoring perfect cubes)
• 13) (-32)^{2/5} (real or not? simplify if real)
• 14) (-16)^{3/4} (real or not? explain briefly)

C. Mixed operations and careful parentheses

• 15) \frac{x^{1/2} \cdot x^{-3}}{x^{-1/2}}
• 16) \left(x^{-2}y^{1/3}\right)^3
• 17) \frac{(4x)^{1/2}}{x^{-1/2}} (simplify; include restriction)
• 18) \left(\frac{a^{2/3}}{a^{-1/3}}\right)^{-2} (include restriction)
• 19) Compare and simplify: -x^{1/2} versus (-x)^{1/2} (state when each is real)
• 20) Simplify and state restrictions: \frac{1}{\sqrt{x}} + \frac{1}{x^{-1/2}}