Exponent laws

Rational exponents

Upper and lower bounds

MYP Extended Mathematics Rational Exponents Notes

Rational Exponents Are Another Way to Write Roots

A unit fraction is a fraction with numerator 1 (for example, $\frac{1}{2}$ or $\frac{1}{7}$).

Unit fractions appear naturally when we write roots as exponents.

Definition

Rational exponent

An exponent of the form $\frac{m}{n}$ where $m$ and $n$ are integers and $n\neq 0$.

For any positive real number $x$, and integers $m$ and $n$ with $n\neq 0$: $$x^{\frac{m}{n}}=(\sqrt[n]{x})^m=\sqrt[n]{x^m}$$

This statement contains three equivalent ways to see the same number or expression:

$x^{\frac{m}{n}}$ is exponent form.
$(\sqrt[n]{x})^m$ means "take the $n$th root, then raise to the power $m$".
$\sqrt[n]{x^m}$ means "raise to the power $m$, then take the $n$th root".

Note

The restriction "$x$ is a positive real number" is not just a detail.
It prevents ambiguity and non-real results when even roots of negative numbers appear (explained later).

The Key Link: Roots Correspond to Unit Fractions

The most basic rational exponents are unit fractions: $$x^{\frac{1}{n}}=\sqrt[n]{x}$$

Then you can build other rational exponents by applying a further integer power: $$x^{\frac{m}{n}}=\left(x^{\frac{1}{n}}\right)^m$$

Analogy

Think of $x^{\frac{m}{n}}$ as a two-step machine: "root" is set by the denominator $n$, and "power" is set by the numerator $m$.

A Reliable Method to Evaluate Expressions Like $a^{m/n}$

To evaluate a number with a rational exponent, it helps to separate the exponent into a unit fraction and an integer.

Example question

Simplify $16^{\frac{3}{4}}$ without a calculator.

Solution

Split the exponent: $$16^{\frac{3}{4}}=\left(16^{\frac{1}{4}}\right)^3$$

Now evaluate the fourth root: $$16^{\frac{1}{4}}=\sqrt[4]{16}=2$$

So: $$16^{\frac{3}{4}}=2^3=8$$

You could also do it in the other order: $$16^{\frac{3}{4}}=\left(16^3\right)^{\frac{1}{4}}=\sqrt[4]{4096}=8$$

Tip

When the base is a perfect power (like $16=2^4$), "take the root first" is usually faster.
When the exponent makes the power convenient first, "power first" can be easier.
Both are valid as long as you stay consistent.

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Note

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Tip

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Rational exponents (also called fractional indices) let you write radicals using exponent notation and then use the familiar laws of exponents to simplify expressions.

Rational Exponents Are Another Way to Write Roots

A unit fraction is a fraction with numerator 1 (for example, $\frac{1}{2}$ or $\frac{1}{7}$).
Unit fractions appear naturally when we write roots as exponents.

Definition

Rational exponent

An exponent of the form $\frac{m}{n}$ where $m$ and $n$ are integers and $n\neq 0$.

For any positive real number $x$, and integers $m$ and $n$ with $n\neq 0$: $$x^{\frac{m}{n}}=(\sqrt[n]{x})^m=\sqrt[n]{x^m}$$
This statement contains three equivalent ways to see the same number or expression:
1. $x^{\frac{m}{n}}$ is exponent form.
2. $(\sqrt[n]{x})^m$ means "take the $n$th root, then raise to the power $m$".
3. $\sqrt[n]{x^m}$ means "raise to the power $m$, then take the $n$th root".

Note

The restriction "$x$ is a positive real number" is not just a detail.
It prevents ambiguity and non-real results when even roots of negative numbers appear (explained later).

The Key Link: Roots Correspond to Unit Fractions

The most basic rational exponents are unit fractions: $$x^{\frac{1}{n}}=\sqrt[n]{x}$$
Then you can build other rational exponents by applying a further integer power: $$x^{\frac{m}{n}}=\left(x^{\frac{1}{n}}\right)^m$$

Analogy

Think of $x^{\frac{m}{n}}$ as a two-step machine: "root" is set by the denominator $n$, and "power" is set by the numerator $m$.

A Reliable Method to Evaluate Expressions Like $a^{m/n}$

To evaluate a number with a rational exponent, it helps to separate the exponent into a unit fraction and an integer.

Example question

Simplify $16^{\frac{3}{4}}$ without a calculator.

Solution

Split the exponent: $$16^{\frac{3}{4}}=\left(16^{\frac{1}{4}}\right)^3$$

Now evaluate the fourth root: $$16^{\frac{1}{4}}=\sqrt[4]{16}=2$$

So: $$16^{\frac{3}{4}}=2^3=8$$

You could also do it in the other order: $$16^{\frac{3}{4}}=\left(16^3\right)^{\frac{1}{4}}=\sqrt[4]{4096}=8$$

Tip

When the base is a perfect power (like $16=2^4$), "take the root first" is usually faster.
When the exponent makes the power convenient first, "power first" can be easier.
Both are valid as long as you stay consistent.

Negative Rational Exponents Mean 'Reciprocal' First

A negative exponent means take the reciprocal: $$x^{-k}=\frac{1}{x^k} \quad (x\neq 0)$$
The same idea works for rational exponents: $$x^{-\frac{m}{n}}=\frac{1}{x^{\frac{m}{n}}}$$

MYP Extended Mathematics Rational Exponents Notes

Rational Exponents Are Another Way to Write Roots

The Key Link: Roots Correspond to Unit Fractions

A Reliable Method to Evaluate Expressions Like $a^{m/n}$

Negative Rational Exponents Mean 'Reciprocal' First

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Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

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Rational Exponents Are Another Way to Write Roots

The Key Link: Roots Correspond to Unit Fractions

A Reliable Method to Evaluate Expressions Like $a^{m/n}$

Negative Rational Exponents Mean 'Reciprocal' First

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Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

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Advanced Number and Bounds3 subtopics

Inequalities and Modelling1 subtopic

Function Transformations4 subtopics

Advanced Probability1 subtopic

Geometric Transformations1 subtopic

Variation and Standard Deviation3 subtopics

Trigonometric Functions and Rules4 subtopics

Sequences, Rational Expressions and Equations5 subtopics

Covariance and Correlation3 subtopics

Logarithms and Non-Linear Inequalities3 subtopics

Graph Theory and Vectors4 subtopics

MYP Extended Mathematics Rational Exponents Notes

Advanced Number and Bounds3 subtopics

Inequalities and Modelling1 subtopic

Function Transformations4 subtopics

Advanced Probability1 subtopic

Geometric Transformations1 subtopic

Variation and Standard Deviation3 subtopics

Trigonometric Functions and Rules4 subtopics

Sequences, Rational Expressions and Equations5 subtopics

Covariance and Correlation3 subtopics

Logarithms and Non-Linear Inequalities3 subtopics

Graph Theory and Vectors4 subtopics

Rational Exponents Are Another Way to Write Roots

The Key Link: Roots Correspond to Unit Fractions

A Reliable Method to Evaluate Expressions Like $a^{m/n}$

Negative Rational Exponents Mean 'Reciprocal' First

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account

Rational Exponents Are Another Way to Write Roots

The Key Link: Roots Correspond to Unit Fractions

A Reliable Method to Evaluate Expressions Like $a^{m/n}$

Negative Rational Exponents Mean 'Reciprocal' First

anim nostrud sit dolore minim proident quis fugiat velit et eiusmod nulla quis nulla mollit dolor sunt culpa aliqua

Duis aute irure dolor in reprehenderit

Excepteur sint occaecat cupidatat non proident

Unlock the rest of this chapter with a free account