Everything I Know About Exponents


2. Describe how powers represent repeated multiplication.

The exponent of a number shows how many times this number is multiplied by itself.                                             



4. Demonstrate the difference between two given powers in which the exponent and the base are interchanged by using repeated multiplication, such as two to the power of three and three to the power of 2.

The difference is that we have a different base multiplied by itself a different number of times. For example, 23 = 2 x 2 x 2 = 8 and 32 = 3 x 3 = 9. In most cases, the products will be different values except, when one of the numbers is 2 and the other is 4. 24 = 2 x 2 x 2 x 2 = 16 and 42 = 4 x 4 = 16

6. Explain the role of parentheses in powers by evaluating a given set of powers such as (-2)4, (-24) and -24.

The position of the parentheses determines if a negative sign will be affected by an even exponent. For example,

(-2)4 = (-2)(-2)(-2)(-2) = 16

(-24) = -1 x 2 x 2 x 2 x 2 = -16

-24 = -1 x 2 x 2 x 2 x 2 = -16

8. Explain the exponent laws for raising a product and quotient to an exponent.

The product rule is when you multiply two exponents with the same base, you keep the base and add the powers together. If there are any coefficients, you have to multiply them. The quotient rule is when you divide two exponents with the same base, you keep the base and subtract the exponents. If there are coefficients, you have to divide them if possible or write them as a simplified fraction.

10. Use patterns to show that a power with an exponent of zero is equal to one.

A power with an exponent of zero is always equal to one except if the base is 0. This comes from the quotient law of exponents because…

12. Use patterns to explain the negative exponent law.

The negative exponent is equal to the reciprocal of the base raised to the same, but positive power.

14. I can identify the error in a simplification of an expression involving powers.

Applying the exponent laws for product, quotient, and power, I can easily find the error in an exponent expression that has been simplified.

16. Determine the sum and difference of two powers.

First, you have to calculate each power individually and then add or subtract.

The sum and difference of two powers is a very powerful factoring technique. It is used a lot in mathematics as part of theorems, like the Pythagorean theorem, or to simplify radicals. Popular sums and differences of powers are:

a3 + b3 = (a + b)(a2 + ab + b2)

a2 – b2 = (a – b)(a + b)

18. Use powers to solve measurement problems.

We use powers to solve for the area of a square, circles, or equilateral triangles and volumes for cubes, cylinders and more! Some of the formulas for area and volume use one of the measurements of the shape as a power like the side length of the radius. Those formulas are for regular polygons and regular 3-D shapes. Knowing the laws of exponents will help you solve the problem and avoid some common errors. In one of the examples below, where there is a square inside a square, a common error can be 62 – 32 = (6-3)2 = 32 = 9, not 27.

20. Applying the order of operations on expressions with powers involving negative exponents and variable bases.

When evaluating problems with exponents, we still use the same order of operations as you would with any other problem. First, you have to always simplify what is inside the brackets. Preferably, you would convert the powers with negative exponents first to prevent confusion and errors, although it is not mandatory. When having powers with different bases, you would first simplify and calculate them individually before you do the operation between them. I wrote two solutions. They are both correct but the alternate solution doesn’t use complex fractions and is simpler.






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