Exponent Laws MPM2D: Principles of Mathematics Consider the - - PDF document

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MPM2D: Principles of Mathematics

Exponent Laws

• J. Garvin

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Exponent Laws

Consider the expression x2 · x3. Using the definition of exponentiation, x2 · x3 can be expressed as (x · x)(x · x · x) = x · x · x · x · x = x5. More generally, xa · xb = (x · x · . . . · x)

• a times

· (x · x · . . . · x)

• b times

= x · x · . . . · x

• a+b times

= xa+b.

Product of Like Powers Law

For any real, non-zero values a, b and x, xa · xb = xa+b. If the bases are not the same, this rule does not apply. The expression 24 · 32 cannot be simplified further.

• J. Garvin — Exponent Laws

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Exponent Laws

Next, consider the expression x3 x2 . Rewriting, x3 x2 can be expressed as x · x · x x · x = x. More generally, xa xb = (x · x · . . . · x)

• a times

(x · x · . . . · x)

• b times

= x · x · . . . · x

• a-b times

= xa−b.

Quotient of Like Powers Law

For any real, non-zero values a, b and x, xa xb = xa−b. Like the earlier Product Law, the bases must be the same.

• J. Garvin — Exponent Laws

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Exponent Laws

Now, consider

• x32.

Rewriting,

• x32 becomes (x · x · x) · (x · x · x) = x6.

In general, (xa)b = (x · x · . . . · x

• a times

) · (x · x · . . . · x

• a times

) · . . . · (x · x · . . . · x

• a times

)

• b times

= xab.

Power of a Power Law

For any real, non-zero values a, b and x, (xa)b = xab.

• J. Garvin — Exponent Laws

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Exponent Laws

Consider (xy)2 next. In its longer form, (xy)2 = (xy)(xy) = (x · x)(y · y) = x2y2. In general, (xy)a = (xy) · (xy) · . . . · (xy)

• a times

= (x · x · . . . · x)

• a times

· (y · y · . . . · y)

• a times

= xaya.

Power of a Product Law

For any real, non-zero values a, x and y, (xy)a = xaya.

• J. Garvin — Exponent Laws

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Exponent Laws

Like the power of a product, the power of a quotient can be similarly defined. For instance, x y 2 = x y

• ·

x y

• = x2

y2 . In general, x y a = x y

• ·

x y

• · . . . ·

x y

• a times

= xa ya

Power of a Quotient Law

For any real, non-zero values a, x and y, x y a = xa ya .

• J. Garvin — Exponent Laws

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Exponent Laws

Example

Simply the expressions x4 · x7, z8 z6 ,

• k35, (2p)3 and

x 2 5 . x4 · x7 = x4+7 = x11. z8 z6 = z8−6 = z2.

• k35 = k3×5 = k15.

(2p)3 = 23p3 = 8p3. x 2 5 = x5 25 = x5 32.

• J. Garvin — Exponent Laws

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Exponent Laws

What about the expression x0? According to the Quotient Law, xa xa = xa−a = x0. At the same time, k k = 1, as long as k = 0. If k = xa, then k k = xa xa = x0 = 1.

Zero Exponent Law

For any real, non-zero value of x, x0 = 1.

• J. Garvin — Exponent Laws

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Exponent Laws

What does a negative exponent, like x−2, mean? Since x x3 = x1−3 = x−2, and since x x3 = 1 x2 , then x−2 = 1 x2 . In general, xa · x−a = xa+(−a) = x0 = 1, assuming x = 0. Therefore, xa · x−a = 1, which can be rearranged to x−a = 1 xa .

Negative Exponent Law

For any real, non-zero value of x and any real, positive value

• f a, x−a = 1

xa .

• J. Garvin — Exponent Laws

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Exponent Laws

Example

Evaluate 1 234 5670. Since the base is non-zero, 1 234 5670 = 1.

Example

Express x−4 using positive exponents. x−4 = 1 x4 . Again, x cannot equal zero.

• J. Garvin — Exponent Laws

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Exponent Laws

Sometimes it is necessary to combine two or more exponent laws to simplify an expression.

Example

Simplify x5y3 x2y7 , using positive exponents. x5y3 x2y7 = x5−2y3−7 = x3y−4 = x3 y4

• J. Garvin — Exponent Laws

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Exponent Laws

Example

Simplify (5p−3q)−2, using positive exponents. (5p−3q)−2 = 5−2p(−3)(−2)q−2 = 1 52 · p6 · 1 q2 = p6 25q2

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Scientific Notation

Scientific notation is a system, used in many sciences, that expresses numbers using powers of 10. For example, the number 352 can be expressed as 3.52 × 102, since 3.52 × 102 = 3.52 × 100 = 352. It is often used as a shorthand notation for very small or very large numbers. For instance, 3 800 000 000 000 (3 trillion, 800 billion) can be expressed more simply as 3.8 × 1012. By convention, scientific notation expresses all numbers with

• ne digit before the decimal point – that is, 4.3 × 103 rather

than 43 × 102. Positive exponents indicate the decimal point has been shifted left, while negative exponents indicate a right shift.

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Scientific Notation

Example

Express 75 328 143 using scientific notation, to two decimal places. Shifting the decimal 7 places to the left, and rounding down, 75 328 143 = 7.53 × 107.

Example

Express 0.000 031 874 using scientific notation, to two decimal places. Shifting the decimal 5 places to the right, and rounding up, 0.000 031 874 = 3.19 × 10−5.

• J. Garvin — Exponent Laws

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Questions?

• J. Garvin — Exponent Laws

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