Roots and Exponents

Squaring Decimals

(.000005)^2, 5^2 = 25, .000005 has 6 decimal places so 6 x 2 = 12, the answer will have 12 decimal places.

Multiplication of like bases

(x^a)(x^b) = x^a+b, keep the based and add the exponents

Multiplication of Different Bases and Like Exponents

(x^a)(y^a) = (xy^a) When multiplying different bases that share a common exponent, keep the common exponent and multiply the bases. i.e. (2^4)(3^4) = 6^4

Zero raised to any positive power is 0

0^n = 0. 0^2 = 0

The cube root

A positive perfect cube is a number other than 1 whose prime factorization contains only exponents that are multiples of 3. First 11 cube roots = 0,1,8,27,64,125,216,343,512,729;1,000

Add and subtract only like radicals

Add and subtract like radicals only. Two or more radicals are alike if they both have the same root index and the same radicand (expression under the radical). i.e. 30 x the square root of 2 can be added to 15 x the square root of 2

Exponents apply to each constituent in their parentheses

An exponent can be distributed evenly over only multiplication or division (4abc)^2 = 4^2 x a^2 x b^2 x c^2 (5/7)^2 = 5^2/7^2 = 25/49

Bases raised to the first power

Any based raised to the first power is that base.

Prime factorization helps us simplify fractional expression with exponents

As a general rule, we try to simplify expressions and calculations as much as possible.

Exponential notation can be factored

Be on the lookout for ways to factor values written in exponential notation. 50^100 = 50 x 50 x 50^98

Be careful when squaring a binomial

Binomial = addition or subtraction of two terms. When squaring a binomial, multiply the binomial by itself using the FOIL process. (a+b)^2 = (a+b)(a+b) = a^2 + b^2 + 2ab

Perfect Square

Consists of only even exponents. 0,1,2,4,8,9,16,25,36,49,64,81,100,121,144,169,196,225

Approximating other roots

Figure out what roots the number is nearest to in order to estimate what it would reduce down to. i.e. The 4th root of 80 = ?, we know that 2^4 is 16 and 3^4 is 81 so the number must be close to 3.

Which is bigger when x is a fraction, x^2, the square root of x or x.

If 0 < x < 1, it must be true that x^2 < x < the square root of x.

Exponents

If A doesn't equal 0, 1 or -1, and a^x = a^y, the x=y. If a^x x a^y = a^z, then x+y=z. We can't say this for 1 because 1 raised to any power is 1.

The Square Root

If n is even, nth root of x^n = |x|, if n is odd, nth root of x^n = x

There is a single term radical in the denominator

If we havee an answer of 3/the square root of 5, we have to rationalize it to get the square root of 5 out of the denominator. If A is not a perfect square, to rationalize a one-term radical, the square root of A, in the denominator of a fraction, multiply that fraction by the square root of A/the square root of A.

Solving equations with square roots

If we see an unknown that needs to be squared, we must isolate the square root first.

Any Nonzero base raised to the zero power equals one

If x doesn't equal 0, x^0 = 1. Anything raised to 0 equals 1.

Comparing Radicals and Exponents

If x, y and m are positive, then x>y if and only if x^m > y^m. Raise expressions to the LCD to compare fraction sizes and then compare final numbers. i.e. Is the 4th root of 4 greater than the 5th root of 7? Turn both into exponents so 4^1/4 and 7^1/5. The LCD is 20 so 4^20/4 = 4^5. 7^20/5 = 7^4. 1024<2401 so the 5th root of 7 is bigger.

Adding and subtracting fractions that contain exponents

Make sure the denominators are the same before adding or subtracting

Approximating Square Roots

Memorize these non perfect square roots: - square root of 2 = 1.4 - square root of 3 = 1.7 - square root of 5 = 2.2 - square root of 6 = 2.4 - square root of 7 = 2.6 - square root of 8 = 2.8 Also try to approximate square roots based on what the number is closest to

Radicals in the Denominator

On the GMAT, radicals must be removed from the denominator in order for the expression to be considered simplified.

Taking the square root of an expression or an equation

On the gmat the square root of a number is always positive. The square root of a variable squared is always the absolute value of that variable. In other words, the square root of x^2 = |x|.

Base is a positive proper fraction and exponent is a positive proper fraction

Result is larger. (1/4)^1/2 = The square root of 1/4 = 1/2

Rule: Base is greater than 1 and exponent is a positive proper fraction

Result is smaller because we are taking the root of a number. 4^1/2 = square root of 4 = 2

Multiple Square Roots

The a root of the b root of x = (x^1/b)^1/a = x^1/b x x^1/a = x^1/ab See example qs for this chapter around the square root of a square root of a square root.

Cube roots of large and small perfect cubes

The cube root of a perfect cube integer has exactly 1/3 the number of zeros to the right of the final nonzero digit as the original perfect cube. i.e. the cubed root of 1,000,000 = 100 the cubed root of .000027 = .03

Solving equations with the variable raised to an even power

The even indexed root of any number is always positive. The 4th root of 81 is 3. If n is even, the nth root of a variable raised to the nth power is equal to the absolute value of that variable. In other words, the nth root of x^n = |x|. Is means x is X digits away from 0 on the number line so the answer could be positive or negative.

Dividing Radicals

The nth root of a / the nth root of b = the nth root of a/b

Addition and subtraction of radicals

The order of operations stipulates that we perform any addition or subtraction under a radical prior to taking the root. The square root of a+b does not equal the square root of a + the square root of b i.e. the square root of 16 + 9 = the square root of 25, so it equals 5

Rule: Base is less than -1 and exponent is an even positive integer.

The result is larger (-4)^2 = 16

Rule: Base is a negative proper fraction and exponent is an odd positive integer greater than 1

The result is larger because it becomes a smaller negative number. (-1/4)^3 = -1/64

Rule: Base is a negative proper fraction and exponent is an even positive integer

The result is larger. (-1/4)^2 = 1/16

Rule: Base is greater than 1 and exponent is an odd positive integer greater than 1

The result is larger. x>1, n>1 and n is odd ---> x^n > x. 4^3 = 64

Rule for figuring out whether the new number is greater than or less than the original number: Base is greater than 1 and exponent is an even positive integer

The result is larger. x>1, n^0 and n is even ---> x^n > x. 4^2 = 16

Rule: Base is less than -1 and exponent is an odd positive integer

The result is smaller. (-4)^3 = -64

Rule: Base is a positive proper fraction and exponent is an even positive integer

The result is smaller. (1/4)^2 = 1/16

Rule: Base is a positive proper fraction and exponent is an odd positive integer greater than 1

The result is smaller. (1/4)^3 = 1/64

Radicals can be expressed in exponential form

The square root of a value is equivalent to raising the value to the 1/2 power. The square root of x = x^1/2, the cubed root of x = x^1/3. In general, for any positive number x, b root of x^a = x^a/b. The square root of x times the square root of x equals x.

Estimating with exponents

Two numbers with the same base and exponents that differ by as little as 1 can be vastly different from each other. This difference is especially pronounced when the bases and exponents are relatively large. example: Approximate the value of 9^8 - 2^12. Is the approximate value closer to 9^8 or 9^7? 2^12 is not very large compared to 9^8, thus the approximate value is much closer to 9^8.

Removing the Radicals with the LCD of the indices of the Radicals

We can remove an exponent attached to a number by raising each expression to the least common denominator i.e. solve for x. x^1/60 = 2^1/10. (x^1/60)^60 = (2^1/10)^60. x = 2^6, x = 64

Prime factorization with exponents

When a nonprime base it raised to an exponent, the expression can be reduced through prime factorization i.e. 6^80 = (3 x 2)^80 = 3^80 x 2^80

Square roots of large and small perfect squares

When a perfect square ends with an even number of zeros, the square root of such a perfect square will have exactly half the number of zeroes to the right of the final nonzero digits as the perfect square. i.e. the square root of 10,000 is 100 If a decimal with a finite number of decimal places is a perfect square, its square root will have exactly half the number of decimal places. i.e. the square root of .16 = .4

Scientific notation

When a positive niumber n is written in scientific notation, it's written in the form of a x 10^b where a is a number between 1 and 10 and b is an integer. B is the number of decimal places. i.e. 4.04 x 10^-6 = .000004

Addition and subtraction of like bases or like radicals

When adding or subtracting expressions with exponents, consider factoring out common factors. For example, x^4 + x^4 + x^4 + x^4 = x^4(1+1+1+1) = x^4(4). When adding or subtracting expressions with radicals, we should also consider factoring out common radical factors. Special trick: 3^n + 3^n + 3^n = 3^n(1+1+1) = 3^n(3) = 3n^n+1

If the bases are not the same, attempt to make them the same

When bases appear not to be equal, reduce the bases using factorization in an attempt to uncover any common bases that may exist in the Q.

Division of Different Bases and Like Exponents

When dividing different bases that share a common exponent, keep the common exponent and divide the bases. if y doesn't equal 0, x^a/y^a = (x/y)^a

Division of like bases

When dividing like bases, keep the like base and subtract the exponents. x^a/x^b = x^a-b

Multiplication and Division with scientific notation

When multiplying and dividing numbers in scientific notation, the coefficients can be multiplied or divided separately from the powers of ten. i.e. (3.5 x 10^5) x (40 x 10^6) = (3.5 x 40) x (10^5 x 10^6)

Multiplying and dividing non radicals and radicals

When multiplying or dividing expression that combine non radicals and radicals, multiply and divide non-radicals by non-radicals, and radicals by radicals (if the radicals have the same index). (a x the nth root of b)(c x the nth root of d) = ac x the nth root of bd (a x the nth root of b)/(c x the nth root of d) = a/b x the nth root of b/d

Simplifying Radicals

When simplifying radical expressions, it's often helpful to first locate and simplify any perfect squares or perfect cubes within the expression. i.e. What is the value of the square root of 200 + the square root of 50? Answer is 15xthe square root of 2.

Be careful when taking the square root of a binomial squared

When taking the square root of a squared binomial, the square root of (x+y)^2 = |x+y|. The applies when you take the nth root of any binomial raised to the nth power.

Power To a Power Rule

When there are exponents that are then being raised to a power, multiply these exponents (x^a)^b = x^ab

Comparing fractions with exponents

When we need to compare fractions with different denominators, one of the ways to do this is to convert them into fractions with a common denominator.

The denominator has two terms (a binomial) and one or both of those terms is a radical

a-b is a binomial and the conjugate of this is a+b (just switch the signs). The conjugate of a-the square root of b is a+the square root of b. (a+b) x (a-b) = a^2-b^2 so the product of a conjugate pair is the difference of squares. We use the conjugate rule to simplify binomials in the form of a+the square root of b, a-the square root of b in the denominator of a fraction.

Raising a base to a negative exponent

if x doesn't equal 0, x^-1 = 1/x and in general, x^-y = 1/x^y. When a fraction is raised to a negative exponent, flip the fraction and make the exponent positive. If x doesn't equal 0 and y doesn't equal 0, then (x/y)^-2 = (y/x)^2 x ^-1/2 = 1/x^1/2 = 1/the square root of x. But we can't have a square root on bottom so must multiply top and bottom by square root of x.

Multiplying Radicals

the mth root of a x the mth root of b = the mth root of a x b. The mth root of a x b = the mth root of a x the mth root of b. If m is even, a and b are nonnegative. If m is different a and b, they cannot be combined.

Factoring out the GCF in expressions containing powers of variables

x^2+y^4 + x^3y^2 can be factored to x^2y^2(y^2+x). Look for the smallest power of x in both and factor. If the terms have numerical coefficients, we will factor out the GCF of these numerical coefficients.