In mathematics, raising a number to a power refers to multiplying a number by itself a certain number of times. The number that is multiplied is called the base, and the number of times it is multiplied is called the exponent or power.
For example, 52 means 5 x 5 = 25. Here, 5 is the base and 2 is the exponent. Similarly, 53 means 5 x 5 x 5 = 125, where 5 is still the base but the exponent is now 3.
Quick Answer
When a number is raised to the power of 1, the result is the original number itself. So 5 raised to the 1st power is simply 5.
This is because any number raised to the 1st power is equal to itself. Mathematically, this can be written as:
x1 = x
Where x is any real number. Substituting 5 for x gives:
51 = 5
Therefore, the answer is that 5 raised to the 1st power is equal to 5.
Explanation
To understand why a number raised to the 1st power equals itself, we need to look at what exponents actually mean.
An exponent tells you how many times to multiply a base number by itself. For example:
- 23 means 2 x 2 x 2 = 8
- 34 means 3 x 3 x 3 x 3 = 81
- 45 means 4 x 4 x 4 x 4 x 4 = 1024
So an exponent of 1 simply means to take the base and multiply it by itself 1 time. Multiplying a number by itself once just gives you back the original number.
Formal Definition
Formally, raising a number x to the power of 1 is defined as:
x1 = x * 1 = x
Where:
- x is the base (any real number)
- 1 is the exponent
- Multiplying any number by 1 returns the original number
So by definition, any base raised to the 1st power equals the base itself.
Examples
Let’s look at some examples to build intuition:
- 21 = 2 * 1 = 2
- 51 = 5 * 1 = 5
- (-3)1 = -3 * 1 = -3
- π1 = π * 1 = π
This shows that regardless of the base, when raised to the 1st power it equals itself. This applies to all real numbers and mathematical constants.
Special Cases
There are some special cases worth noting:
Zero to the Power of One
01 = 0 * 1 = 0
Zero raised to any power is still zero.
One to the Power of One
11 = 1 * 1 = 1
One raised to any power is still one.
Proof by Induction
We can prove rigorously using mathematical induction that x1 = x for all x:
- Let P(n) be the statement that n1 = n
- Base Case: Let n = 1. Then 11 = 1. So P(1) is true.
- Inductive Step: Assume P(k) is true for some k, that is k1 = k.
- Then for k+1, (k+1)1 = (k+1)*1 = k+1.
Therefore, if P(k) is true, P(k+1) is also true. - By mathematical induction, P(n) is true for all natural numbers n.
- Since this holds for all natural numbers, which includes 0, 1, 2, etc., this proof holds for all real numbers x.
- Therefore, by the principle of mathematical induction, x1 = x is true for all real x.
This inductive proof formally verifies that any number raised to the power of 1 equals itself.
Graphical Interpretation
We can also show this graphically:
This graph shows y=xn for different values of n. When n=1, the function is just the line y=x. This line has a slope of 1 and passes through the origin, showing that x1=x.
Real World Applications
Some real world examples where numbers are raised to the 1st power include:
- Finance: Simple interest calculations involve raising the principal amount to the 1st power to get the total back.
- Geometry: The length, width, and height dimensions of a cube raised to the 1st power give the original dimensions.
- Programming: Looping or iterating through a block of code once involves raising a counter variable to the 1st power in each iteration.
Exponentiation Rules
Understanding that a number raised to the 1st power equals itself is key to memorizing rules of exponents such as:
- Product Rule: xm * xn = xm+n
- Quotient Rule: xm / xn = xm-n
- Power Rule: (xm)n = xm*n
In all these formulas, we use the fact that x1=x in various algebraic manipulations and proofs.
Related Exponent Concepts
Understanding exponents of 1 is related to grasping other exponent concepts:
- Zero Exponents: x0 = 1 for any non-zero x
- Negative Exponents: x-n = 1/xn
- Fractional Exponents: x1/n is the nth root of x
So a solid foundation in the basic principle that x1 = x is essential for mastering exponent rules and advanced exponent concepts in algebra, calculus and beyond.
Common Mistakes
Some common mistakes when working with exponents of 1 include:
- Thinking x1 = 1 for all x. This is wrong, as x1 = x.
- Forgetting that x1 = x works for negative bases too. (-5)1 = -5, not 1.
- Misusing x1 = x to incorrectly simplify exponents, e.g. thinking 32 * 34 = 37.
Being aware of these common errors can help avoid making similar mistakes when working through exponent problems.
Conclusion
In summary, when any base is raised to the 1st power, the result is just the base number itself. This comes from the definition of exponents, where x1 means to multiply x by itself 1 time, which gives back x. This fundamental rule that x1 = x holds true for all real number bases and is essential in simplifying and manipulating exponent expressions in algebra, calculus and other areas of mathematics.
Practice Problems
Here are some practice problems to test your understanding of exponents of 1:
- Simplify: 41
- Simplify: (-2)1
- Simplify: (x+3)1
- Simplify: (2y)1
- Simplify: 34 * 31
Solutions:
- 41 = 4
- (-2)1 = -2
- (x+3)1 = x+3
- (2y)1 = 2y
- 34 * 31 = 81 * 3 = 243
Being able to quickly simplify exponents of 1 will help in evaluating more complex exponential expressions down the line.
Frequently Asked Questions
Why is a number raised to the power of 1 equal to itself?
A number raised to the 1st power is equal to itself because an exponent of 1 means to multiply the base number by itself 1 time. Multiplying any number by itself once returns the original number. So x1 = x*1 = x for all real x.
Does 01 = 0? Why?
Yes, 01 = 0. This is because 0 raised to any power is still 0. When the base number is 0, multiplying it by itself any number of times (including 1 time for an exponent of 1) the result is still 0.
Is 11 = 1 true? Why?
Yes, 11 = 1. This is because 1 raised to any exponent equals 1. So raising 1 to the 1st power yields 1 * 1 = 1. This shows that the rule x1 = x holds even when x = 1.
Can you raise a negative number to the power of 1?
Yes, negative numbers can also be raised to the 1st power. For example (-3)1 = -3. The rule works for all real number bases, whether positive, negative, fractions or decimals.
What are some common mistakes when working with exponents of 1?
Common mistakes include incorrectly thinking x1 = 1 for all x, forgetting the rule works for negative bases, and misapplying the rule to incorrectly simplify exponents in algebraic expressions. Being aware of these will help avoid similar errors.
Key Takeaways
- Raising a number x to the 1st power gives back the original number: x1 = x.
- This comes from the definition of exponents, where x1 means multiply x by itself 1 time.
- This rule holds true for all real number bases x, including negatives and fractions.
- Understanding this basic exponent rule is key for learning more advanced exponent properties and manipulations.
| Base | Raised to Power of 1 |
|---|---|
| 5 | 5 |
| 0 | 0 |
| -2 | -2 |
| π | π |
This table summarizes some examples of bases raised to the 1st power, showing the result is always the original base.
Conclusion
In summary, raising any base to the 1st power gives back the base itself. This fundamental rule underlies exponent properties and manipulations studied in advanced mathematics. Understanding that x1 = x provides a foundation for working with exponents that is essential for algebra, calculus, and beyond.