Exponent Rules

After addition, subtraction, multiplication, and division, exponentiation serves as the 5th and final arithmetic operation.

Calculations involving exponents are crucial in algebra and are a major feature of SAT/ACT math. Seven basic rules and two additional corollaries govern exponentiation.

It’s important to understand these principles well and master their use through practice and application.

Below are the laws governing exponents you'll need to know and follow.


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Basic Exponent Rules (A ≠ 0, B ≠ 0)

Product of Equal-Base Powers: A^m*A^n = A^(m+n)
c: z^3*z^4 = z^7. 

Quotient of Equal-Base Powers: A^m/A^n = A^(m–n).
For example: x^-3/x^5 = x^-8.

Power of a Power: (A^m)^n = A^(mn)
For example: (y^3)^4 = y^12.

Power of a Product: (A*B)^n = (A^n)(B^n)
For example: (x^2*y)^3 = (x^6)(y^3).

Power of a Quotient: (A/B)^n = [(A)^n]/[(B)^n]
For example: (x^7.5/y^-2)^2 = [x^15]/[ y^-4].

Zero Powers: A^0 = 1 (A ≠ 0)
For example: (2z–1)^0 = 1 (z ≠ 1/2).

Negative Powers: A^-n = 1/(A^n)
For example: x^-3 = 1/(x^3).

Additional Corollaries

Quotient of Negative Powers: A^-m/B^-n = B^n/A^m
For example: y^-1/z^4 = z^-4/y^1.
(Changing positions of the lower and/or upper powers changes the signs on those exponents)

Negative Power of a Quotient: (A/B)^-n = (B/A)^n
For example: (x^-1/y^8)^-4 = (y^8/x^-1)^4.

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For practice, search Google for “exponent rules worksheet,” pick a worksheet that provides answers, complete the worksheet, analyze any mistakes, and redo it until you can complete that worksheet with no errors. Then repeat, with additional worksheets, as needed, until you’ve mastered this material.

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Contact Chris

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

General Functions – What You Need to Know

Much of high school algebra revolves around the study of input/output machines called functions, one of the most widely applicable concepts in all mathematics. Naturally, functions comprise a large fraction of questions found on the SAT/ACT. Fortunately, only knowledge of basic facts and processes is required.

Here’s what you need to know, generally, about functions. 

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[Note: “iff” means “if and only if.”]

Definition

A function is a relationship between two sets of numbers, one containing inputs and the other for outputs; these sets are called "the domain" and "the range," respectively. A function can can be understood as an input/output “machine” that takes a number in and returns a corresponding number out, such that no input is associated with more than one output. Normally, the input is called x and the output is called y. The function itself is named with a single letter, like f, in which case the output for general input x can be written “f(x),” pronounced “f of x.”

y = f(x) 
y and f(x) are interchangeable. 

Function values
The “value of a function” is an output value (y value).

Operations
The essential operation with functions is substitution.
“g(n)” means substitute n for x in function g.

Composition of Functions
Composite functions are “nested” functions. “f[g(x)]” means function g is nested inside function f.
For example: To find f[g(2)], first find g(2) and then substitute that value into f. 

Zeros of a function
Values of x (input values) that make y (output values) equal zero.
Zeros are found at x-intercepts.

When f(x) = 0, solutions are called “roots.”

Solutions iff roots iff zeros iff x-intercepts (“roots,” “zeros,” “solutions,” “x-intercepts” are essentially synonymous).

Intercepts of functions:
To find intercepts, let the other variable’s value be zero

For example: For the y-intercept, let x = 0).

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For practice, search Google for worksheets covering any or all of the above, pick a worksheet that provides answers, complete the worksheet, analyze any mistakes, and redo it until you can complete that worksheet with no errors. Then repeat, with additional worksheets, as needed, until you’ve mastered this material.

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Contact Chris

borlandeducational.com

Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Unit Conversions

Kilometers to centimeters? Gallons to tablespoons? Feet-per-second squared to miles-per-hour squared? 

Unit conversion is a pre-algebra topic that stops many students in their tracks. Questions about converting units pop up routinely on the SAT/ACT.

Basic conversions are easy to calculate by simple multiplication or division. More difficult problems require “Dimensional Analysis,” an easy and reliable way to perform conversion calculations.

The method is based on the following facts:

1. Equations relating units enable the creation of two fractions whose values are 1;

2. Multiplication by 1 never changes values and is therefore always allowed; 

3. "Per" implies division.  

To see how they enable the conversion of units using Dimensional Analysis, let's look at two questions.

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Simple example

1 mile = 5280 feet. Therefore, 1 mi / 5280 ft and 5280 ft / 1 mi are two fractions with values = 1. Let's convert 45 miles into feet. First write, as a fraction, the quantity to be converted: 45 miles / 1, and then multiply by 1 in the form of 5280 ft / 1 mi (we choose this fraction, with miles below, in order to cancel-out miles). Cancelling “mi” above and below leaves “ft” as the unit and 45 * 5280 as the calculation. So the answer is 237,600 ft.

Complex example

We'll convert 3500 meters per second squared to kilometers per hour squared. First write, as a fraction, the quantity to be converted: 3500 m/s^2. Since 1 kilometer = 1000 meters and 1 hour = 3600 seconds, multiply the initial fraction by 1 in the following forms: 1 km / 1000 m, 3600 s / 1 hr, and 3600 s / 1hr (to cancel s^2 below). Cancelling above and below leaves “m/ hr^2” as the unit and 3500 * 3600 * 3600 / 1000 as the calculation. So the answer is 45,360,000 km/hr^2.

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For practice, search Google for “converting units dimensional analysis worksheet,” pick a worksheet that provides answers, complete the worksheet, analyze any mistakes, and redo it until you can complete that worksheet with no errors. Then repeat, with additional worksheets, as needed, until you’ve mastered this material.

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Contact Chris

borlandeducational.com

Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Seeing is Believing

On the SAT/ACT, test takers are warned figures aren’t necessarily drawn to scale. In recent years, however, questions with misshapen diagrams have become vanishingly rare.

Nowadays, unless a drawing is clearly distorted, students can assume all figures to be scale drawings. And from this can be inferred a tremendously helpful geometry strategy.

Based on the realism of figures drawn to scale, the notion that “seeing is believing” can be used to make good estimates helpful in answering even the most irksome questions.

For example: Angles that seem equal probably are equal. Lines look parallel? Call it true. If one segment appears to be slightly less than half the length of another, that can be assumed.

Known information in geometric figures can thus be used to “ball park” reasonable guesses about unknown information in the same figure, and this is often all it takes to find the correct answer or at least eliminate wildly incorrect ones.

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Contact Chris

borlandeducational.com

Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Transforming Functions

Questions involving reflecting or shifting graphs stump a great many students.

But this needn’t be! 

Four simple rules govern all transformation questions encountered on the SAT/ACT. Master these laws, and all such questions suddenly become easy ones.

Listed below is what you need to know.

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Horizontal Reflection (across the y-axis)

Replace x with -x. 

For example: If f(x) = x^2– x+1, the horizontal reflection is f(x) = (-x)^2–(-x)+1 = x^2+x+1.

Vertical Reflection (across the x-axis)

Replace y with -y.
For example: If g(x) = 3x–2 i.e. y = 3x–2, the vertical reflection is (-y) = 3x–2 and y = -3x+2. Therefore, g(x) = -3x+2.

Horizontal Shift, h units

Replace x with x–h.
For example: If f(x) = x^2–x is shifted 4 units left, h = -4, h–k = h–(-4) = h+4, and the shifted function is f(x) = (x+4)^2–(x+4) = x^2+8x+16–x–4 = x^2+7x+12.

Vertical Shift, k units

Replace y with y-k (or simply add k to the function).
For example: If y = |6x–1| is shifted 3 units up, k = 3, y–k = y–3, and the shifted function is y–3 = |6x–1|. Therefore, y |6x–1|+3.

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For practice, search Google for worksheets covering any or all topics listed above, pick a worksheet that provides answers, complete the worksheet, analyze any mistakes, and redo it until you can complete that worksheet with no errors. Then repeat, with additional worksheets, as needed, until you’ve mastered this material.

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Contact Chris

borlandeducational.com

Copyright © 2006-present: Christopher R. Borland. All rights reserved.