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September 6th, 2023
Empowering Mathematics: An Introduction to Powers and Exponents
In the world of mathematics, powers and exponents are fundamental concepts that play a pivotal role in simplifying calculations, representing repeated multiplication, and expressing relationships between numbers. This article provides a clear and accessible introduction to powers and exponents, exploring what they are, how they work, and their practical applications in various mathematical and real-world contexts.
AUTHOR
Mr. Calvin Musk, Chief executive officer at Calvin Industries Corporation and President of Calvin State University
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Learning Intention
I can represent and identify the parts of a power (base and exponent) and write a power as repeated multiplication and in standard form and vice versa
In mathematics, power refers to repeated multiplication, just like how multiplication is repeated addition or how division is repeated subtraction. They are often referred to as exponents. Whether it's unraveling the secrets of compound interest, solving complex equations, or describing the intricacies of exponential growth, powers serve as a versatile tool that empowers mathematicians and scientists alike to explore.
Exponents and powers can be used to represent extremely big or extremely small numbers in a more straightforward fashion. To illustrate 3 x 3 x 3 x 3 in a straightforward manner, for instance, we could write it as 34, where 4 is the exponent and 3 is the base.
Parts of a Power
Before we can begin discussions regarding powers, we must first understand the parts of power and some important terminology regarding powers.
Image Source : Calvin State University, Faculty of Mathematics
Exponents and powers can be used to represent extremely big or extremely small numbers in a more straightforward fashion. To illustrate 3 x 3 x 3 x 3 in a straightforward manner, for instance, we could write it as 34, where 4 is the exponent and 3 is the base.The base is the number that is raised to a certain power. In the expression "a^n," "a" is the base. The exponent represents the number of times the base is multiplied by itself. In the expression "a^n," "n" is the exponent. A power is the result of raising a base to an exponent. For example, in "2^3," 2 raised to the power of 3 equals 8. On a computer, we typically can represent powers and exponents through the ^ symbol to indicate a raised number.
Important Terminology
Squared: To square a number means to raise it to the power of 2. For example, 5^2 is read as "5 squared" and equals 25.
Cubed: To cube a number means to raise it to the power of 3. For example, 4^3 is read as "4 cubed" and equals 64.
Square Root: The square root of a number "a" is another number "b" such that when "b" is squared, it equals "a." It is denoted as √a.
Radical: A radical is a symbol (√) used to represent the square root or other roots of a number. For example, √9 represents the square root of 9, which is 3.
Exponential Form: Exponential form is a way of expressing a number using a base and an exponent. For example, 10^4 is the exponential form of 10,000.
Negative Exponent: A negative exponent indicates the reciprocal of a number raised to a positive exponent. For example, 2^(-3) is equal to 1/(2^3), which is 1/8.
Zero Exponent: Any non-zero number raised to the power of 0 is equal to 1. For example, a^0 = 1, where "a" is any non-zero number.
Fractional Exponent: A fractional exponent represents a root of a number. For instance, a^(1/2) represents the square root of "a," and a^(1/3) represents the cube root of "a."
Power of Ten Notation: This notation is commonly used to express very large or very small numbers using powers of 10. For example, 3.2 × 10^5 represents 320,000.
Scientific Notation: Scientific notation is a way of expressing numbers as a product of a number between 1 and 10 and a power of 10. For example, 6.02 × 10^23 represents Avogadro's number.
Exponential Growth: Exponential growth refers to a pattern of growth where the quantity multiplies by a fixed factor over equal intervals of time.
Exponential Decay: Exponential decay is the opposite of exponential growth, where a quantity decreases by a fixed factor over equal intervals of time.
Exponents and powers can be used to represent extremely big or extremely small numbers in a more straightforward fashion. To illustrate 3 x 3 x 3 x 3 in a straightforward manner, for instance, we could write it as 34, where 4 is the exponent and 3 is the base.The base is the number that is raised to a certain power. In the expression "a^n," "a" is the base. The exponent represents the number of times the base is multiplied by itself. In the expression "a^n," "n" is the exponent. A power is the result of raising a base to an exponent. For example, in "2^3," 2 raised to the power of 3 equals 8. On a computer, we typically can represent powers and exponents through the ^ symbol to indicate a raised number.
Integer Rules in Multiplication
When you multiply 2 positive integers ( + ) x ( + ), you will always get a positive integer as the product. You can imagine this as repeated addition. You have a number and you are adding it x amount of times.
Example ) (+3) x (+4) = (+12)
When you multiply 2 negative integers ( - ) x ( - ), you will always get a positive integer as the product. You can imagine this as removing x amount of groups from a pre-existing group.
Example ) (-3) × (-4) = +12
When you multiply a positive integer by a negative integer or a negative integer by a positive integer, the product will always be negative. You can imagine this as increasing the amount of groups by times x of a pre-existing group.
Example ) (-3) x (+4) = -12
We can summarize this as the product of two numbers with the same sign is positive and where the product of two numbers with different signs is negative.
General Practice
Image Source : Calvin State University, Faculty of Mathematics
Using exponents on a negative integer can be a little bit different than using exponents on a positive integer. But, it's the same process. We take the given value and multiply it by itself x amount of times as indicated by the exponent. Here are some examples on how they should look.
On the top right, we are first calculating what's shown in the bracket. In the bracket, we are given the values 4 exponent 3. Doing the calculations, we get 4 x 4 = 16 and 16 x 4 which gets us a product of 64. Once we raise 4 to the power of 3, we then apply the negative sign of the product to get -64.
On the bottom left, we are given two negative symbols. Two negatives cancel each other out to form a positive integer, meaning that we are left with 4 to the power of 3. Doing the appropriate calculations, we get 4 x 4 = 16 x 4 = +64 as the product.
On the bottom right, the same principle applies. We have two negative signs which cancel out to form a positive four, and we multiply four by itself (4 to the power of 2) to get the final product as 16.
Predicting Positives or Negatives
To determine whether or not a product will be positive or negative based on two given integers, we can follow these sets of rules:
If the base is positive and the exponent is even, the answer is positive.
If the base is positive and the exponent is odd, the answer is positive.
If the base is negative and the exponent is even, the answer is positive.
If the base is negative and the exponent is odd, the answer is negative.
Here are some examples)
3^2: The base (3) is positive, and the exponent (2) is even. So, the answer is positive.
10^4: The base (10) is positive, and the exponent (4) is even. So, the answer is positive.
8^5: The base (8) is positive, and the exponent (5) is odd. So, the answer is positive.
(-6)^5: The base (-6) is negative, and the exponent (5) is odd. So, the answer is negative.
-(6^5): The base (6) is positive, and the exponent (5) is odd. So, the answer is negative.
(-4)^2: The base (-4) is negative, and the exponent (2) is even. So, the answer is positive.
-(7^3): The base (7) is positive, and the exponent (3) is odd. So, the answer is negative.
-2: The number is already negative, so the answer is negative.
-(6^4): The base (6) is positive, and the exponent (4) is even. So, the answer is negative.
-(-6)^4: The base (-6) is negative, and the exponent (4) is even. So, the answer is positive.
-(-(4^4)): The base (4) is positive, and the exponent (4) is even. So, the answer is positive.
Zero and Negative Exponents
When we take a number and raise it to the power of one, the value remains the same. For example, 10 raised to the power of one basically means that the value remains 10, as any number raised to the power of one is equal to itself. If we take the same value and raise it to the power of two instead, that indicates that we are squaring the number, which means multiplying it by itself. In the case of 10 raised to the power of two, it equals 100, as we are calculating 10 multiplied by 10.
But what happens when we raise it to the power of zero? What happens when we raise it to the power of a negative integer as opposed to a positive one?
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