Tricks for Algebra 2

Tricks with exponents can make algebra less intimidating.

For many people, math and algebra are tedious and intimidating subjects. Long lines of numbers, variables and symbols may appear confusing, and students may dread the thought of spending hours working through problems that seem to have no real-world connections. Knowledge of certain algebraic tricks can help to simplify expressions, and speed up the solution of equations. Many of these tricks involve exponents.

Multiplying Exponents

One important trick to know is how to multiply constants and variables with exponents. If you are multiplying numbers with the same exponent, the product is the product of the bases, to the power of the given exponent. This works regardless of whether the bases are constants or variables, or both. For example, (10x)^2 * 5^2 = (50x)^2, or 2,500x^2. On the other hand, if you are multiplying identical bases with different exponents the product is the same base to the power of the sum of the exponents. For example, (2x)^2 * (2x)^1 = (2x)^3, or 8x^3.

Exponent Identities

In lower levels of math and algebra, the identity properties of zero and one are very helpful because they can be used to simplify expressions. Similar properties apply to higher levels of algebra. Any number to the power of one is equal to that number without any specified exponent. Also, any number to the power of zero is equal to one. In other words, x^1 = x, while x^0 = 1. Knowing these properties can help to simplify equations that might otherwise look very intimidating.

Irrational Denominators

Another trick concerns how to remove a radical from the denominator. If there is no addition or subtraction involved in the denominator of an expression, you can get rid of the radical in the denominator by multiplying both the numerator and the denominator by a radical. For example, if you had the square root of two in the denominator you would multiply both top and bottom by the square root of two, so that the denominator would simplify to two. For example, if you have ther term 1/√2, you would multiply it by √2/√2, yielding the answer √2/2.

Conjugates

If addition or subtraction are involved in the denominator, you can still simplify the expression by multiplying the denominator and the numerator by the conjugate of the expression in the denominator. The conjugate of an expression involving two terms is the same expression with the sign between the two terms changed. The conjugate of x - 5 is x + 5. When an expression and its conjugate are multiplied together, the product is the square of the first term minus the square of the second term. For example, (x – 5) * (x + 5) = x² - 25. If there is a radical in an expression in the denominator, such as 3 - √2, it can be eliminated by multiplying both top and bottom by the conjugate of the denominator.

References

• Purplemath: Exponents- Basic Rules
• Math Is Fun: Rationalize the Denominator