What are Integer Exponents?
Integer exponents, mathematically refer to those exponents that should be integers whether positive integers or negative integers. The positive integer exponents denote the number of times a number should be multiplied by itself. The negative exponents as a rule need to first be flipped and then multiplied.
What are Integers?
Did you know that the word integer is derived from the Latin word integer meaning whole? Integers refer to a number that can be represented as a non-fraction. That is numbers that can be written without a fractional component are defined as an integer.
Net gears can be positive or negative.Examples of integer are- 1, 4, 9, -7, -66, etc.
What are Exponents?
Exponents represent the mathematical operation of exponentiation. Exponents of a number show how many times a number should be used in multiplication whether with itself or with other numbers.
For example, 23, where 3 is the exponent and means that two should be multiplied with itself three times.
Integers and Exponents
Simply put, all integers can be exponents-whether positive or negative. Exponents denote the number of times a base number should be multiplied whether with itself or with another number.
For example, the expression 5 × 5 can also be written as 52. The integer here is both 2 and 5 but the exponent is 2. The exponent 2 here denotes that 5 has to be multiplied twice with itself.
Similarly, 43 stands for 4 × 4 × 4 HD shows that 4 should be multiplied with itself three times.
Integer Exponents Rule
Seven major exponents rules are the answer to the question-“how to solve integer exponents”. These rules are all-encompassing in terms of mathematical operations such as addition, multiplication, division, etc.
Let us understand these rules properly:
1. Product of Powers Rule
When two bases of the same number are to be multiplied, add the exponents while keeping the base number the same.
For example- 32 × 34 = 32+4
= 36
= 3 × 3 × 3 × 3 × 3 × 3
= 729
2. Quotient of Powers Rule
When two bases with the same number are being divided, subtract the exponents while keeping the base the same.
For example- 46 divided by 42 = 46-2
= 44
= 4 × 4 × 4 × 4
= 256
3. Power of a Power Rule
When an exponent is being raised to another exponent, multiple the two exponents and keep the base the same.
For example- (52)3 = 56
= 5 × 5 × 5 × 5 × 5 × 5
= 15,625
4. Power of a Product Rule
When two bases are being multiplied by the same exponent, distribute the exponent to each of the bases.
For example- (41 × 51)2 = 42 × 52
= 16 × 25
= 400
5. Power of a Quotient Rule
When a power is raised to a quotient-distribute it evenly to both the denominator and numerator.
For example- (⅘)2 = 42/52
= 16/25
6. Zero Power Rule
Any base raised to the power of zero is equal to one.
For example- 30 = 1
7. Negativity Content Rule
When any base is raised to a negative exponent, turn the number into a fraction and then make it reciprocal.
For example- 3-2 = 1/32
The idea behind this rule is to convert the negative exponent to make them into positive ones.
How to Solve Integer Exponents?
Solving integer exponents can be a very easy task if the student is clear with the basics of this concept. As we have explained above Integer exponents are exponents that stand for an integer both negative and positive and denote the number of times the base number should be multiplied with itself or with another number.
Seven rules govern the solving process and all of them have been mentioned by us above. It is through the use of these rules that integer exponent questions can be solved.
Let us look at the general steps of solving such questions-
Step 1. Carefully look at the question.
Step 2. Discern which of the seven formulas would be suitable.
Step 3. Use the chosen formula.
Step 4. Write the answer clearly.
Step 5. Recheck the process and the final answer.
The entire process of employing one of the formulas, finding the answer, and then re-checking would make sure that there is no scope for mistakes and if done they can be corrected. Such a strategy would help make you score high marks and avoid silly mistakes.
FAQs on Integers As Exponents
1. What does it mean to have an integer as an exponent?
Having an integer as an exponent indicates how many times a base number is to be multiplied by itself. It can be a positive, negative, or zero integer.
- A positive exponent (like in 5³) means repeated multiplication: 5 × 5 × 5 = 125.
- A zero exponent (like in 5⁰) always results in 1, provided the base is not zero.
- A negative exponent (like in 5⁻³) indicates the reciprocal of the positive exponent: 1 / 5³ = 1/125.
2. What are the main laws for working with integer exponents?
The main laws of exponents are fundamental rules used to simplify expressions. For any non-zero integers 'a' and 'b' and integer exponents 'm' and 'n', the key laws are:
- Product Rule: To multiply powers with the same base, add their exponents: aᵐ × aⁿ = aᵐ⁺ⁿ.
- Quotient Rule: To divide powers with the same base, subtract the exponents: aᵐ / aⁿ = aᵐ⁻ⁿ.
- Power of a Power Rule: To raise a power to another power, multiply the exponents: (aᵐ)ⁿ = aᵐⁿ.
- Zero Exponent Rule: Any non-zero base raised to the power of zero is 1: a⁰ = 1.
- Negative Exponent Rule: A base raised to a negative power is the reciprocal of the base raised to the positive power: a⁻ⁿ = 1/aⁿ.
3. How do you simplify an expression with a negative integer exponent?
To simplify an expression with a negative integer exponent, you need to find its reciprocal. The rule is a⁻ⁿ = 1/aⁿ. For example, to solve 4⁻², you first calculate the base with the positive exponent (4² = 16) and then take its reciprocal, which is 1/16. If the negative exponent is in the denominator (like 1/x⁻³), it moves to the numerator and becomes positive (x³).
4. What is the difference between (-5)² and -5²?
This is a common point of confusion that depends on the order of operations.
- In (-5)², the parentheses indicate that the base is -5. The calculation is (-5) × (-5), which equals a positive 25.
- In -5², there are no parentheses, so the exponent applies only to the 5. The calculation is -(5 × 5), which equals -25. The negative sign is applied after the squaring.
5. Why is any non-zero number raised to the power of zero equal to 1?
This rule can be logically proven using the quotient law of exponents. Consider the expression 7³ / 7³. We can solve this in two ways:
1. Any number divided by itself is 1. So, 7³ / 7³ = 1.
2. Using the quotient rule (aᵐ / aⁿ = aᵐ⁻ⁿ), we get 7³⁻³ = 7⁰.
Since both methods must yield the same answer, it logically follows that 7⁰ = 1. This holds true for any non-zero base.
6. How are integer exponents used in real-life applications?
Integer exponents are crucial in many fields for handling very large or very small quantities. Key examples include:
- Scientific Notation: Scientists use positive exponents to express vast distances (e.g., distance to a star) and negative exponents to describe minuscule sizes (e.g., the diameter of an atom).
- Computer Technology: Data storage is measured in powers of 2, like kilobytes (2¹⁰ bytes) and megabytes (2²⁰ bytes).
- Finance: Compound interest calculations use exponents to determine the growth of investments over time.
- Biology: They are used to model population growth (positive exponents) and the rate of radioactive decay (negative exponents).
7. How do negative exponents help in writing very small numbers in standard form?
Standard form (or scientific notation) makes it easy to write and compare very small decimals. A negative exponent is used to show how many places the decimal point has been moved to the right. For instance, the number 0.000058 can be written in standard form as 5.8 × 10⁻⁵. The -5 exponent is a concise way of showing that the original number is very small and that the decimal point was shifted 5 places.
8. Can the base of an expression with an integer exponent be a fraction?
Yes, absolutely. The laws of exponents are not limited to integer bases; they apply to any real number, including fractions and decimals. When a fraction is raised to a power, both the numerator and the denominator are raised to that power. For example, (2/3)⁴ = 2⁴ / 3⁴ = 16/81. Similarly, for a negative exponent, you take the reciprocal of the base first: (2/3)⁻⁴ = (3/2)⁴ = 81/16.
