INTRODUCTION
Everything covered in this lesson, in one place - useful for revision or printing.
\( a^m \times a^n = a^{m+n} \)
Exponents are a short way of writing repeated multiplication: \( 2^5 = 2\times2\times2\times2\times2 \).
By the end of this lesson you will be able to:
• use the laws of exponents
• simplify expressions by writing every base as a prime
• work with rational exponents
• solve exponential equations by making the bases equal
• handle equations where you must factorise a common power first
\( a^m \times a^n = a^{m+n} \) \( \dfrac{a^m}{a^n} = a^{m-n} \)
When the bases are the same:
• multiplying: add the exponents
• dividing: subtract the exponents
The bases must match. \( 2^3 \times 3^2 \) cannot be combined into one power.
\( (a^m)^n = a^{mn} \) \( (ab)^n = a^n b^n \)
A power raised to a power: multiply the exponents.
A product raised to a power: the exponent goes to every factor inside the bracket. This is the law that lets you split a base such as 18 into its prime factors later on.
\( a^0 = 1 \) \( a^{-n} = \dfrac{1}{a^n} \)
• Anything (except 0) to the power 0 is 1.
• A negative exponent means reciprocal: move the power across the fraction line and the sign of the exponent changes.
These laws hold for \( a > 0 \), which is the case for every base in this lesson.
\( y^{10}x^{-2} \)
You are given \( y^5 = 50 \) and \( x^2 = 10 \). Find the value of \( y^{10}x^{-2} \).
Do not try to find x and y. Rewrite the expression using the powers you were given:
\( \text{same base only} \)
• \( a^m \times a^n = a^{m+n} \) and \( a^m \div a^n = a^{m-n} \)
• \( (a^m)^n = a^{mn} \) and \( (ab)^n = a^nb^n \)
• \( a^0 = 1 \) and \( a^{-n} = \dfrac{1}{a^n} \)
Watch out for: combining powers with different bases.
\( 18 = 2 \times 3^2 \)
The laws only combine powers with the same base. An expression with bases 18, 6 and 36 looks impossible, but every one of them is built from the primes 2 and 3.
So the first move in any simplification is: write every base as a product of primes.
\( 36 = 2^2 \times 3^2 \)
The composite bases used in this lesson:
\( 18^{m-2} = 2^{m-2}\cdot 3^{2m-4} \)
Once a base is split, use \( (ab)^n = a^nb^n \) and \( (a^m)^n = a^{mn} \). The exponent multiplies every exponent inside, including any bracket:
\( \text{primes first} \)
• Write each base as a product of primes.
• Push the exponent onto each prime factor.
• Multiply out brackets in the exponents carefully.
Next, you will use this to collapse large expressions to a single number.
\( \dfrac{18^{m-2}\cdot 6^{-m}\cdot 36^{m+2}}{3^{3m}\cdot 2^{2m}} \)
Simplify the expression on the board.
Step 1: write every base as primes.
\( 2^{2m+2-2m}\cdot 3^{3m-3m} \)
Step 2: collect the powers of 2 and the powers of 3. Add the exponents in the numerator, then subtract the exponents in the denominator.
\( 2^2 \cdot 3^0 = 4 \)
Step 3: finish.
\( \dfrac{2^{-2n}\cdot 3^{-3n}}{2^{2n}\cdot 4^{n-1}\cdot 12^{-3n}} \)
Simplify the expression on the board.
Only 4 and 12 need splitting:
\( \dfrac{3^{2x-1}\cdot 5^{x-3}}{45^{x-2}} \)
Quick check: Simplify the expression on the board. Split 45 into primes first.
\( \text{primes, collect, simplify} \)
1. Write every base as primes.
2. Collect the exponents of each prime separately (add on top, subtract the bottom).
3. Simplify; use \( a^0 = 1 \) and \( a^{-n} = \dfrac{1}{a^n} \).
Watch out for: sign errors when subtracting a bracket such as \( -(2x-4) \).
\( a^{\frac{m}{n}} = \sqrt[n]{a^m} \)
An exponent can be a fraction. The denominator is the root, the numerator is the power. For example \( a^{\frac{1}{n}} = \sqrt[n]{a} \).
All the laws you already know still work with fractional exponents.
\( 2^{0{,}2} = 2^{\frac{1}{5}} = \sqrt[5]{2} \)
A decimal exponent is a fraction in disguise: \( 0{,}2 = \dfrac{1}{5} \). Convert it when you need to see which root it is.
Often you will not need the root at all: keep the power in the form \( 2^{0{,}2} \) and use \( (a^m)^n = a^{mn} \).
\( M = 2^{0{,}2},\; M^b = 16 \)
Quick check: If \( M = 2^{0{,}2} \) and \( M^b = 16 \), find the value of b.
\( a^{\frac{m}{n}} = \sqrt[n]{a^m} \)
• Denominator = root, numerator = power.
• Change decimal exponents to fractions when helpful.
• The power-of-a-power law, \( (a^m)^n = a^{mn} \), does most of the work.
\( a^x = a^y \;\Rightarrow\; x = y \)
In an exponential equation the unknown is in the exponent.
Method:
1. Isolate the power.
2. Write both sides as a power of the same base.
3. Equate the exponents and solve.
(This works for a base \( a > 0,\; a \neq 1 \).)
\( 5^{2x-1} - 1 = 0 \)
Solve for x. The trick: \( 1 = 5^0 \).
\( 11\cdot 3^{2x+1} = 297 \)
Solve for x. Isolate the power before doing anything else: divide by 11.
\( 3^x - 1 = 0 \)
Quick check: Solve for x.
\( 9^{2x+3} = 27^{x+5} \)
Quick check: Solve for x. Write both sides as powers of 3.
\( \text{isolate, same base, equate} \)
• Isolate the power first (move constants, divide out coefficients).
• Remember \( 1 = a^0 \).
• Write both sides with the same prime base, then equate exponents.
\( 5^{x-4} + 5^x = 626 \)
Here two powers are added. There is no law for \( a^m + a^n \), so you cannot combine them directly.
Instead, split each power so that a common factor appears:
\( 5^x\left(5^{-4} + 1\right) = 626 \)
Take out \( 5^x \) as a common factor:
\( x = 4 \)
Now isolate the power and equate exponents:
\( a^x\cdot a^{k} + a^x = a^x(a^k + 1) \)
• Addition or subtraction of powers: factorise.
• Take out the power that contains the variable.
• The bracket becomes a plain number; then isolate the power and equate exponents as before.
\( \text{primes, laws, same base} \)
• Laws: add, subtract and multiply exponents only when the bases match.
• Simplifying: write every base as primes, then collect each prime.
• Rational exponents: denominator = root, numerator = power.
• Equations: isolate the power, make the bases equal, equate the exponents.
• Sums of powers: factorise a common power first.