Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview
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Chapter 9
Worked example (no skipped steps)
9.1 Simplify 25.
25 = 2 · 2 · 2 · 2 · 2 = 32
Answer: 32.
Step-by-step solutions
Worked example (no skipped steps)
9.2 Simplify 𝑥 3 · 𝑥 4.
Same base, so add exponents:
𝑥 3 · 𝑥 4 = 𝑥 3+4 = 𝑥 7
Answer: 𝑥 7.
Worked example (no skipped steps)
9.3 Simplify 𝑦 5 .
𝑦 2
Same base in a quotient, so subtract exponents:
𝑦 5 = 𝑦 5−2 = 𝑦 3
𝑦 2
Answer: 𝑦 3 (with 𝑦 ̸= 0).
Worked example (no skipped steps)
9.4 Simplify ( 𝑎 2)3.
A power raised to a power means multiply the exponents:
( 𝑎 2)3 = 𝑎 2·3 = 𝑎 6
Answer: 𝑎 6.
Worked example (no skipped steps)
9.5 Rewrite without negative exponents: 𝑚−2.
A negative exponent means reciprocal:
1
𝑚−2 = 𝑚 2
Answer:
1 .
𝑚 2
Worked example (no skipped steps)
9.6 Rewrite without negative exponents:
1 .
𝑝−3
Since 𝑝−3 = 1 ,
𝑝 3
1
1
=
= 𝑝 3
𝑝−3
1 /𝑝 3
Answer: 𝑝 3.
171
Worked example (no skipped steps)
√
9.7 Simplify
49.
Because 72 = 49,
√49 = 7
Answer: 7.
Worked example (no skipped steps)
√
9.8 Simplify
72.
Factor 72 as 36 · 2:
√
√
72 =
36 · 2
Now pull out the square root of 36:
√
√
√
√
72 =
36 2 = 6 2
√
Answer: 6 2.
Worked example (no skipped steps)
9.9 Simplify 𝑥 2 · 𝑥 5.
Add exponents:
𝑥 2 · 𝑥 5 = 𝑥 2+5 = 𝑥 7
Answer: 𝑥 7.
Worked example (no skipped steps)
9.10 Simplify 𝑦 7 .
𝑦 3
Subtract exponents:
𝑦 7 = 𝑦 7−3 = 𝑦 4
𝑦 3
Answer: 𝑦 4 (with 𝑦 ̸= 0).
Worked example (no skipped steps)
9.11 Rewrite 𝑎−3 with a positive exponent.
A negative exponent means reciprocal:
1
𝑎−3 = 𝑎 3
Answer: 1 .
𝑎 3
Step-by-step solutions
Worked example (no skipped steps)
9.12 Simplify (2 𝑎 3)(3 𝑎 2).
Multiply the coefficients:
2 · 3 = 6
Now add exponents on 𝑎:
𝑎 3 · 𝑎 2 = 𝑎 3+2 = 𝑎 5
So:
(2 𝑎 3)(3 𝑎 2) = 6 𝑎 5
Answer: 6 𝑎 5.
Worked example (no skipped steps)
9.13 Rewrite with positive exponents: 𝑥−2.
1
𝑥−2 = 𝑥 2
Answer: 1 .
𝑥 2
Worked example (no skipped steps)
9.14 Simplify (3 𝑏 2)2.
Square the coefficient and square the variable factor:
(3 𝑏 2)2 = 32( 𝑏 2)2 = 9 𝑏 4
Answer: 9 𝑏 4.
Worked example (no skipped steps)
9.15 Simplify 𝑐 5 .
𝑐 5
Subtract exponents:
𝑐 5 = 𝑐 5−5 = 𝑐 0
𝑐 5
Any nonzero number to the zero power is 1, so:
𝑐 0 = 1
Answer: 1 (with 𝑐 ̸= 0).
173
Worked example (no skipped steps)
9.16 Simplify ( 𝑚 3 𝑛 2)( 𝑚 2 𝑛).
Combine like bases:
𝑚 3 · 𝑚 2 = 𝑚 5
𝑛 2 · 𝑛 = 𝑛 3
So:
( 𝑚 3 𝑛 2)( 𝑚 2 𝑛) = 𝑚 5 𝑛 3
Answer: 𝑚 5 𝑛 3.
Worked example (no skipped steps)
9.17 Simplify 32·34 .
33
Combine the powers in the numerator:
32 · 34 = 36
Now divide:
36 = 36−3 = 33 = 27
33
Answer: 27.
Worked example (no skipped steps)
√
9.18 Simplify
27.
Factor 27 as 9 · 3:
√
√
27 =
9 · 3
Pull out the square root of 9:
√
√
27 = 3 3
√
Answer: 3 3.
Worked example (no skipped steps)
√
9.19 Simplify
50.
Factor 50 as 25 · 2:
√
√
50 =
25 · 2
Pull out the square root of 25:
√
√
50 = 5 2
√
Answer: 5 2.
Step-by-step solutions
Worked example (no skipped steps)
9.20 Rewrite without negative exponents: 4−1.
A negative exponent means reciprocal:
1
4−1 = 4
Answer: 1 .
4