Practice Makes Perfect Algebra - Carolyn Wheater [20]
Put each expression in simplest radical form. Assume all variables are positive numbers.
Rationalizing denominators
If the denominator of an expression is a radical or a multiple of a radical, you can remove the radical, or rationalize the denominator, by multiplying the numerator and denominator of the fraction by the radical in the denominator. This is the same method you use to express fractions with a common denominator. Multiplying both numerator and denominator by the same number is equivalent to multiplying by 1, so it changes the appearance of the fraction but not the value.
If the denominator is a sum or difference that includes a radical, you will still want to multiply the numerator and denominator by the same number, but multiplying by just the radical will not be effective. You will need to multiply by the conjugate of the denominator, the same two terms connected by the opposite sign. The conjugate of is , and the conjugate of is . Multiplying by the conjugate eliminates the radical because when you multiply the sum and difference of the same two terms using the FOIL rule, the middle terms, where the radicals would have been, add to 0.
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EXERCISE 8.2
Rationalize the denominators and put each expression in simplest radical form.
Adding and subtracting radicals
Only like radicals can be combined by addition or subtraction, and they combine like variable terms, by adding or subtracting the coefficients, the numbers in front. , but cannot be combined.
Problems that look like unlike radicals at first glance may simplify down to like radicals. When you need to combine radicals by addition or subtraction, first put each term in simplest radical form and then combine like radicals.
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EXERCISE 8.3
Simplify each expression as completely as possible.
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Solving radical equations
Radical equations are equations that contain one or more radicals with the variable in the radicand. The key to solving radical equations is isolating the radical and then raising both sides of the equation to a power so that the radical sign is lifted.
One radical
To solve an equation containing one radical, isolate the radical by moving all terms that do not involve the radical to the other side. Square both sides of the equation, and solve the resulting equation. Be sure to check solutions in the original equation to eliminate any extraneous solutions.
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EXERCISE 8.4
Solve each equation. Check your answers in the original equations. If the equation cannot be solved, write No solution.
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Two radicals
When an equation contains two radicals, choose one to eliminate first. Isolate that radical, and square both sides. This may mean that you need to FOIL one side, and you’ll probably find that while squaring the isolated radical eliminates that radical, FOIL multiplication on the other side will lift one radical sign but introduce another. Just isolate the remaining radical, square both sides again, and solve the resulting equation. Be sure to check solutions in the original equation. Extraneous solutions are common.
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EXERCISE 8.5
Solve each equation. Check for extraneous solutions.
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Graphing square root equations
The graph of a square root equation looks like half of a parabola lying on its side. The graph of begins at the origin and forms a slowly rising curve in the first quadrant as shown in Figure 8.1. A negative multiplier in front of the radical will flip the graph over the x-axis.
Figure 8.1 The top half of a parabola on its side is the graph of the square root function.
Because negative numbers have no square roots in the real numbers, the graph only exists for values of x that make the radicand positive. The graph of begins at(h, 0), and the graph of begins at(−h, 0). A constant added on the end of the equation moves the graph up or down. Consider where the graph begins before choosing x-values for your table of values.
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EXERCISE 8.6
Graph each function by making a table of values and plotting points.