Power Rules & Problem Solving

 

Understanding the Power Rules

The power rules are a set of guidelines that describe how we deal with different kinds of powers and roots. It is important to understand and learn these so you can spot them in questions!

Key Power Rules

 

Power Rule	Description
$a^m \times a^n = a^{m+n}$	When multiplying like terms, add their exponents.
$a^m ÷ a^n = a^{m-n} \text{or} \frac{a^m}{a^n} =  a^{m-n}$	When dividing like terms, subtract the exponents
$(a^m)^n = a^{mn}$	When raising a power to another power, multiply the exponents
$(ab)^n = a^n \times b^n$	Distribute the power to each term inside the parentheses.
\(\eft(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \)	Distribute the power to both the numerator and the denominator.
$a^0 = 1$	Anything (except 0) raised to the zero power equals one.
$a^{-1} = \frac{1}{a} // a^{-n} = \frac{1}{a^n}$	A negative exponent indicates reciprocal.
$a^1 = a$	Anything raised to the power 1 equals the same thing.
$(ab)^n = a^nb^n$	Raising two terms multiplied together to a power is the same as raising them individually and multiplying
$\left(\frac{a}{b}\right)^{-1} = \frac{b}{a}$  $\left(\frac{a}{b}\right)^{-n} = \frac{b^n}{a^n}$	Raising a fraction to a negative power will flip/inverse the fraction
$a^{ \frac{1}{n} } = \sqrt[n]{a}$	Raising anything to a fractional power is the same as having that thing to the root of $n$. Where n is the denominator.
$a^{ -\frac{1}{n} } = \left(a^{ \frac{1}{n} } \right)^{-1}$ $\left(\sqrt[n]{a}\right)^{-1} = \frac{1}{\sqrt[n]{a}}$	Anything raised to a negative fractional power is the same as 1 over the root $n$
$a^{ \frac{m}{n} } = (a^{ \frac{1}{n} \times m}$ $= \left(a^{ \frac{1}{n} } \right) = (\sqrt[n]{a})^m$ $= (a^m)^{\frac{1}{n} } = \sqrt[n]{a^m}$	A number raised to a fractional power is the same as rooting to the denominator and raising to the power of the numerator

 

Using Power Rules in Algebraic Problem Solving

Let's look at some worked examples that apply these rules.

Example 1: Multiplying Powers

Problem: Simplify $x^3 \times x^4$

Solution:

• Use the rule for multiplying powers with the same base: add the exponents. $$x^3 \times x^4 = x^{3+4} = x^7$$

 

Example 2: Dividing Powers

Problem: Simplify $\frac{y^6}{y^2}$

Solution:

• Use the rule for dividing powers with the same base: subtract the exponents. $$\frac{y^6}{y^2} = y^{6-2} = y^4$$

Example 3: Power of a Power

Problem: Simplify $(z^3)^2$

Solution:

• Use the power of a power rule: multiply the exponents. $$(z^3)^2 = z^{3 \times 2} = z^6$$

 

Example 4: Power of a Product

Problem: Simplify $(3x)^2$

Solution:

• Use the power of a product rule: apply the power to both the coefficient and the variable. $$(3x)^2 = 3^2 \times x^2 = 9x^2$$

 

Example 5: Negative Exponent

Problem: Simplify $x^{-4}$

Solution:

• Use the negative power rule: flip the base to make the exponent positive. $$x^{-4} = \frac{1}{x^4}$$

 

Example 6: Fractional Exponent

Problem: Simplify \(16^{\frac{1}{2}\)

Solution:

• A fractional exponent means taking the root. Here, $\frac{1}{2}$ is the square root. $$16^{\frac{1}{2}} = \sqrt{16} = 4$$

 

Example

Problem: Write $\frac{1}{\sqrt[4]{x^5} }$ in the form $x^n$ and find the value of $n$

Solution:

Using the rule $a^{\frac{1}{m} } = \sqrt[m]{a}$ the square root turns into the power $\frac{1}{4}$

$$\therefore \frac{1}{\sqrt[4]{x^5} } = \frac{1}{ (x^5)^{\frac{1}{4} } }$$

Now we can you the rule $(a^p)^q = a^{pq}$ to simplify the denomintor of the fraction. So it becomes

$$\frac{1}{ (x^5)^{\frac{1}{4} } } =  \frac{1}{ x^{\frac{5}{4} } }$$

 

 

 

Solving Equations with Variables in the Power

 

Key Steps for Solving Power Equations

1. Rewrite the equation to get the bases on each side of the equation to match (if possible).
2. Solve for the variable by isolating

 

Example: Same Bases on Both Sides

Problem: Solve for $x$ in $2^{3x} = 16$

Solution:

1. Rewrite $16$ as a power of $2$: $$16 = 2^4 \\ \text{So the equation becomes:} \\ 2^{3x} = 2^4$$
2. Since the bases are the same, you can set the exponents equal to each other: $$3x = 4$$
3. Solve for $x$: $$x = \frac{4}{3}$$

 

Example: More Complex Powers

Problem: Solve for $x$ in $3^{x + 2} = 27^{x - 1}$

Solution:

1. Rewrite $27$ as a power of $3$: $$27 = 3^3 \\ \text{So,} \\ 3^{x + 2} = (3^3)^{x - 1}$$
2. Apply the power of a power rule on the right side: $$3^{x + 2} = 3^{3(x - 1)}$$
3. Since the bases are the same, set the exponents equal to each other: $$x + 2 = 3(x - 1)$$
4. Expand and solve for $x$: $$x + 2 = 3x - 3 \\ 5 = 2x \\ x = \frac{5}{2}$$