1.1 Easier algebra with exponents

Mathematics

Grade 9

Numbers

Module 2

Easier algebra with exponents

Easier algebra with exponents

CLASS WORK

• Do you remember how exponents work? Write down the meaning of “three to the power seven”. What is the base? What is the exponent? Can you explain clearly what a power is?
• In this section you will find many numerical examples; use your calculator to work through them to develop confidence in the methods.

1 DEFINITION

2 ³ = 2 × 2 × 2 and a ⁴ = a × a × a × a and b × b × b = b ³

also

(a+b) ³ = (a+b) × (a+b) × (a+b) and ${\left(\frac{2}{3}\right)}^4=\left(\frac{2}{3}\right)\times \left(\frac{2}{3}\right)\times \left(\frac{2}{3}\right)\times \left(\frac{2}{3}\right)$

1.1 Write the following expressions in expanded form:

4 ³ ; (p+2) ⁵ ; a ¹ ; (0,5) ⁷ ; b ² × b ³ ;

1.2 Write these expressions as powers:

7 × 7 × 7 × 7

y × y × y × y × y

–2 × –2 × –2

(x+y) × (x+y) × (x+y) × (x+y)

1.3 Answer without calculating: Is (–7) ⁶ the same as –7 ⁶ ?

• Now use your calculator to check whether they are the same.
• Compare the following pairs, but first guess the answer before using your calculator to see how good your estimate was.

–5 ² and (–5) ² –12 ⁵ and (–12) ⁵ –1 ³ and (–1) ³

• By now you should have a good idea how brackets influence your calculations – write it down carefully to help you remember to use it when the problems become harder.

• The definition is:

a ^r = a × a × a × a × . . . (There must be r a’s, and r must be a natural number)

• It is good time to start memorising the most useful powers:

2 ² = 4; 2 ³ = 8; 2 ⁴ = 16; etc. 3 ² = 9; 3 ³ = 27; 3 ⁴ = 81; etc. 4 ² = 16; 4 ³ = 64; etc.

Most problems with exponents have to be done without a calculator!

2 MULTIPLICATION

• Do you remember that g ³ × g ⁸ = g ¹¹ ? Important words: multiply ; same base

2.1 Simplify: (don’t use expanded form)

7 ⁷ × 7 ⁷

(–2) ⁴ × (–2) ¹³

( ½ ) ¹ × ( ½ ) ² × ( ½ ) ³

(a+b) ^a × (a+b) ^b

• We multiply powers with the same base according to this rule:

a ^x × a ^y = a ^x+y also $a^{x+y}=a^x\times a^y=a^y\times a^x$ , e.g. $8^{\text{14}}=8^4\times 8^{\text{10}}$

3 DIVISION

• $\frac{4^6}{4^2}=4^{6-2}=4^4$ is how it works. Important words: divide ; same base

3.1 Try these: $\frac{a^6}{a^y}$ ; $\frac{3^{\text{23}}}{3^{\text{21}}}$ ; $\frac{{\left(a+b\right)}^p}{{\left(a+b\right)}^{\text{12}}}$ ; $\frac{a^7}{a^7}$

• The rule for dividing powers is: $\frac{a^x}{a^y}=a^{x-y}$ .

Also $a^{x-y}=\frac{a^x}{a^y}$ , e.g. $a^7=\frac{a^{\text{20}}}{a^{\text{13}}}$

4 RAISING A POWER TO A POWER

• e.g. ${\left(3^2\right)}^4$ = $3^{2\times 4}$ = $3^8$ .

4.1 Do the following:

• This is the rule: ${\left(a^x\right)}^y=a^{\text{xy}}$ also $a^{\text{xy}}={\left(a^x\right)}^y={\left(a^y\right)}^x$ , e.g. $6^{\text{18}}={\left(6^6\right)}^3$

5 THE POWER OF A PRODUCT

• This is how it works:

(2a) ³ = (2a) × (2a) × (2a) = 2 × a × 2 × a × 2 × a = 2 × 2 × 2 × a × a × a = 8a ³

• It is usually done in two steps, like this: (2a) ³ = 2 ³ × a ³ = 8a ³

5.1 Do these yourself: (4x) ² ; (ab) ⁶ ; (3 × 2) ⁴ ; ( ½ x) ² ; (a ² b ³ ) ²

• It must be clear to you that the exponent belongs to each factor in the brackets.
• The rule: (ab) ^x = a ^x b ^x also $a^p\times b^p={\left(\text{ab}\right)}^b$ e.g. ${\text{14}}^3={\left(2\times 7\right)}^3=2^3{7}^3$ and $3^2\times 4^2={\left(3\times 4\right)}^2={\text{12}}^2$

6 A POWER OF A FRACTION

• This is much the same as the power of a product. ${\left(\frac{a}{b}\right)}^3=\frac{a^3}{b^3}$

6.1 Do these, but be careful: ${\left(\frac{2}{3}\right)}^p$ ${\left(\frac{\left(-2\right)}{2}\right)}^3$ ${\left(\frac{x^2}{y^3}\right)}^2$ ${\left(\frac{a^{-x}}{b^{-y}}\right)}^{-2}$

• Again, the exponent belongs to both the numerator and the denominator.
• The rule: ${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}$ and $\frac{a^m}{b^m}={\left(\frac{a}{b}\right)}^m$ e.g. ${\left(\frac{2}{3}\right)}^3=\frac{2^3}{3^3}=\frac{8}{\text{27}}$ and $\frac{a^{\mathrm{2x}}}{b^x}=\frac{{\left(a^2\right)}^x}{b^x}={\left(\frac{a^2}{b}\right)}^x$

end of CLASS WORK

TUTORIAL

• Apply the rules together to simplify these expressions without a calculator.

1. $\frac{a^5\times a^7}{a\times a^8}$ 2. $\frac{x^3\times y^4\times x^2{y}^5}{x^4{y}^8}$

3. ${\left(a^2{b}^{3}c\right)}^2\times {\left({\text{ac}}^2\right)}^2\times {\left(\text{bc}\right)}^2$ 4. $a^3\times b^2\times \frac{a^3}{a}\times \frac{b^5}{b^4}\times {\left(\text{ab}\right)}^3$

5. $\left(2\text{xy}\right)\times {\left({\mathrm{2x}}^2{y}^4\right)}^2\times \left(\frac{{\left(x^{2}y\right)}^3}{{\left(2\text{xy}\right)}^3}\right)$ 6. $\frac{2^3\times 2^2\times 2^7}{8\times 4\times 8\times 2\times 8}$

end of TUTORIAL

Some more rules

CLASS WORK

1 Consider this case: $\frac{a^5}{a^3}=a^{5-3}=a^2$