# 1.1 Easier algebra with exponents

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## 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 ${\left(\frac{2}{3}\right)}^{4}=\left(\frac{2}{3}\right)×\left(\frac{2}{3}\right)×\left(\frac{2}{3}\right)×\left(\frac{2}{3}\right)$

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

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. ${\left({3}^{2}\right)}^{4}$ = ${3}^{2×4}$ = ${3}^{8}$ .

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 ${a}^{p}×{b}^{p}={\left(\text{ab}\right)}^{b}$ e.g. ${\text{14}}^{3}={\left(2×7\right)}^{3}={2}^{3}{7}^{3}$ and ${3}^{2}×{4}^{2}={\left(3×4\right)}^{2}={\text{12}}^{2}$

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 $\frac{{a}^{2x}}{{b}^{x}}=\frac{{\left({a}^{2}\right)}^{x}}{{b}^{x}}={\left(\frac{{a}^{2}}{b}\right)}^{x}$

end of CLASS WORK

TUTORIAL

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

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

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

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

end of TUTORIAL

Some more rules

CLASS WORK

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

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