Exercise 40 Page 211

a Any non-zero number raised to the power of zero equals

1 .

Also, a power raised to a negative exponent,

a^{- b},

can be rewritten as

\frac{1}{a ^{b}} .

Knowing this information, we can simplify our expression.

5^{0} \cdot 2^{- 3}

ExponentZero

$a^{0} = 1$

1 \cdot 2^{- 3}

OneMult

$1 \cdot a = a$

2^{- 3}

NegExponentToFrac

$a^{- m} = \frac{1}{a ^{m}}$

\frac{1}{2 ^{3}}

CalcPow

Calculate power

\frac{1}{8}

b Let's simplify the expression by using the rules of multiplying and dividing powers.

\frac{a ^{3}}{b ^{- 2}} \cdot \frac{a b ^{2}}{a ^{4}}

MultFrac

Multiply fractions

\frac{a ^{3} \cdot a b ^{2}}{a ^{4} b ^{- 2}}

MultBasePow

$a \cdot a^{m} = a^{1 + m}$

\frac{a ^{4} b ^{2}}{a ^{4} b ^{- 2}}

ReduceFrac

$\frac{a}{b} = \frac{a / a ^{4}}{b / a ^{4}}$

\frac{b ^{2}}{b ^{- 2}}

DivPow

$\frac{a ^{m}}{a ^{n}} = a^{m - n}$

b^{2 - (- 2)}

SubNeg

$a - (- b) = a + b$

b^{2 + 2}

AddTerms

Add terms

b^{4}

c Multiplication is commutative, which means we can rearrange the factors any way we want without changing the result.

2.3 \times 1 0^{- 3} \times 4.2 \times 1 0^{2}

CommutativePropMult

Commutative Property of Multiplication

2.3 \times 4.2 \times 1 0^{- 3} \times 1 0^{2}

MultPow

$a^{m} \cdot a^{n} = a^{m + n}$

2.3 \times 4.2 \times 1 0^{- 3 + 2}

AddTerms

Add terms

2.3 \times 4.2 \times 1 0^{- 1}

Multiply

9.66 \times 1 0^{- 1}

The expression simplifies to

9.66 \times 1 0^{- 1},

which is in scientific notation.

d Note that any product raised to an exponent can be rewritten by placing the exponent on all factors of the product.

(3.5 \times 1 0^{3})^{2}

PowProd

$(a \cdot b)^{m} = a^{m} \cdot b^{m}$

3 . 5^{2} \times (1 0^{3})^{2}

PowPow

$(a^{m})^{n} = a^{m \cdot n}$

3 . 5^{2} \times 1 0^{3 \times 2}

Multiply

3 . 5^{2} \times 1 0^{6}

CalcPow

Calculate power

12.25 \times 1 0^{6}

Note that

12.25 \times 1 0^{6}

is not in scientific notation. Scientific notation is written as

a \times 1 0^{b},

where

a

is a number between

1

and

10 .

But

12.25

is not between

1

and

10,

so to write it in scientific notation we have to move the decimal point one step to the left and increase the exponent of

1 0^{6}

1 .

12.25 \times 1 0^{6} \Leftrightarrow 1.225 \times 1 0^{7}

Subchapter links

4.1.1 Line of Best Fit p.196-197

4.1.2 Residuals p.201-202

4.1.3 Upper and Lower Bounds p.205-206

4.1.4 Least Squares Regression Line p.210-213