Practice Makes Perfect Algebra - Carolyn Wheater [14]
7x + 7y = 21
2. x + 2y = 7
x + 2y = 9
3. x + y = 11
x − y = − 1
4. 2x + y = 13
8x + 4y = 51
5. 2x − 5y = 3
10x = 15 + 25y
6. 7x + 3y = 24
9x − 3y = 24
7. 11x − 7y = 13
14y = 46 − 22x
8. y = 2x − 3
y = 2x + 4
9. y = 8 − 3x
y = 8 − 5x
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Powers and polynomials
An exponent is a symbol used to show repeated multiplication. The product 5 × 5 × 5 is written 53 to show that 5 is used as a factor 3 times. The expression x4 means x · x · x · x, or the product obtained by using x as a factor 4 times. The number(or variable) that is multiplied is called the base, the little number that tells how many times to use it is the exponent, and together, as in 53 or x4, they form a power.
Rules for exponents
To multiply powers of the same base, keep the base and add the exponents. If you write out in long form what the powers mean, you can see that the result of multiplying powers of the same base is another power of that base, and the new exponent can be found by adding the exponents in the problem.
a2 · a3 =(a · a)(a · a · a) = a · a · a · a · a = a5 = a2+3
If the powers have different bases, there’s really not much you can do. 42 · 33 is the product of two 4s and three 3s. It’s not five of anything. You could evaluate 42 and evaluate 33 and multiply the results, but that won’t work when the bases are variables.
To divide powers of the same base, keep the base and subtract the exponents. Again, if you write out the long form of the problem and cancel, you’ll see that the result is a power of the same base, with an exponent that’s the difference between the two exponents.
To raise a power to a power, keep the base and multiply the exponents. If you say you want to square a power, for example, you’re saying you want to multiply it by itself, to use it as a factor twice. That turns it into a multiplication problem, and you could follow the rule for multiplication. This rule is just a shortcut.
(b3)2 =(b3)(b3) = b3+3 = b6 = b2×3
Special exponents
If you follow the rule for dividing powers to evaluate , you’ll conclude that . You know from arithmetic, however, that any number divided by itself equals 1, so . Put those two ideas together and you get a simple rule: any non-0 number to the 0 power is 1. If a ≠ 0, a0 = 1. Notice that the rule applies only to non-0 numbers. If you tried to do 00, you’d be torn between “any number to the 0 power is 1” and “0 to any power is 0.” The definition of the 0 power comes from dividing, and because division by 0 is undefined, 00 is indeterminate.
Applying the division rule to a problem that has a larger power in the denominator than in the numerator leads to another definition. According to the rules for exponents, and . To understand what these negative exponents mean, write out the problems in their long form.
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EXERCISE 6.1
Simplify each expression.
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More rules
Combining the basic rules for exponents with the associative and distributive properties produces advanced rules for the power of a product and the power of a quotient.
Power of a product
When a product of two or more factors is raised to a power, the associative and commutative properties allow us to find a shortcut. Write out the power, rearrange, and regroup, and you’ll see that each factor in the product is raised to the power.
(4x2y5)3 =(4x2y5)(4x2y5)(4x2y5)
= 4 · 4 · 4 · x2 · x2 · x2 · y5 · y5 · y5
= 43(x2)3(y5)3
When a product is raised to a power, each factor is raised to that power.
Power of a quotient
When a quotient is raised to a power, both the numerator and the denominator are raised to that power.
You can combine the power of a product rule and the power of a quotient rule to handle more complicated expressions, but don’t misapply them. These rules don’t apply to sums or differences.
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EXERCISE 6.2
Simplify each expression.
1.(2x5)2
2.(−2x3)3
3.(5a2)(2a3)2
4.(−x2)(3xy5)3
5.(3b)2(2b3)3
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Monomials and polynomials
A monomial is a single term. Monomials involve only multiplication of