Week 14: Exponent rules and linear systems

Field Book

52 to 55

Due

Test 12 due

Method

Paper first, every step

Two tools this week, and they do not look related until you use them. The power rule tells you what happens when an exponent sits outside a set of parentheses. Substitution tells you how to solve two equations at once by making one of them wear the other's clothes. Both are about replacement: swapping something complicated for something you already know the value of.

What you have to be able to do

Apply the power rule, including when the outside exponent is negative
Solve a system of two equations by substitution
Simplify a complex fraction

The procedures

The power rule

An exponent on the outside of parentheses applies to every factor inside, separately. Multiply the exponents.
If the outside exponent is negative, invert the whole thing first, then apply the positive exponent.
Converting cubed units, use the unit multiplier three times, once for each dimension.

Solving a system by substitution

Solve one equation for one variable. Pick the one that is already almost alone.
Substitute that expression into the other equation. Now there is only one variable left.
Solve it, then put the value back into either original equation to find the second variable.

Complex fractions

A complex fraction is a fraction with fractions inside it.
Multiply the top and the bottom by the LCD of the small fractions, and the small fractions vanish.
Or read the main bar as division and multiply by the reciprocal.

Worked

WORKED

Simplify (2x³)⁴.
The 4 hits both factors: 2⁴ and x^3·4.
ANSWER: 16x¹²

WORKED

Solve the system: y = 3x − 2 and 2x + y = 8.
The first equation already has y alone, so substitute it into the second: 2x + (3x − 2) = 8.
5x − 2 = 8, so 5x = 10 and x = 2. Put x back: y = 3(2) − 2.
ANSWER: x = 2, y = 4

Terms

Term	Meaning
power rule	an exponent outside parentheses multiplies every exponent inside
unit multiplier	a fraction equal to 1 built from two equal measurements, used to convert units
complex fraction	a fraction that has fractions in its numerator or denominator

Check yourself

1. (3x²)³

2. Solve: y = x + 1 and 2x + y = 7

3. Which method would you use on 5x + 2y = 9 when y = 2x − 3?

reveal answers

27x⁶ · x = 2, y = 3 · substitution, y is already alone

This page is the summary, not the lesson. The full working, every step of every example, is on the board in class. Come here to remember what the week was about. Go there to learn how it is done.