# What is 3 to the 6th power?

• Last Updated : 27 Sep, 2021

Mathematics is not only about numbers but it is about dealing with different calculations involving numbers and variables. This is what basically is known as Algebra. Algebra is defined as the representation of calculations involving mathematical expressions that consist of numbers, operators, and variables. Numbers can be from 0 to 9, operators are the mathematical operators like +, -, ×, ÷, exponents, etc, variables like x, y, z, etc.

### Exponents and Powers

Exponents and powers are the basic operators used in mathematical calculations, exponents are used to simplifying the complex calculations involving multiple self multiplications, self multiplications are basically numbers multiplied by themselves. For example, 7 × 7 × 7 × 7 × 7, can be simply written as 75. Here, 7 is the base value and 5 is the exponent and the value is 16807. 11 × 11 × 11, can be written as 113, here, 11 is the base value and 3 is the exponent or power of 11. The value of 113 is 1331.

Exponent is defined as the power given to a number, the number of times it is multiplied by itself. If an expression is written as cxy where c is a constant, c will be the coefficient, x is the base and y is the exponent. If a number say p, is multiplied n times, n will be the exponent of p. It will be written as,

p × p × p × p … n times = pn

### Basic rules of Exponents

There are certain basic rules defined for exponents in order to solve the exponential expressions along with the other mathematical operations, for example, if there are the product of two exponents, it can be simplified to make the calculation easier and is known as product rule, let’s look at some of the basic rules of exponents,

• Product Rule ⇢ an + am = an + m
• Quotient Rule ⇢ an / am = an – m
• Power Rule ⇢ (an)m = an × m or m√an = an/m
• Negative Exponent Rule ⇢ a-m = 1/am
• Zero Rule ⇢ a0 = 1
• One Rule ⇢ a1 = a

### What is 3 to the 6th power?

Solution:

Any number having a power of 6 can be written as the exponent of 6. Say x raised to the power of 6, can be written as x6. The power 6 of a number is the number multiplied by itself six times, a 6th power of the number is represented as the exponent 6 on that number. If a power 6 of x has to be written, it will be x6. For instance, the power 6 of 5 is represented as 56 and is equal to 5 × 5 × 5 × 5 × 5 × 5 = 15625. Another example can be the power 6 of 12, represented as 126, which is equal to 12 × 12 × 12 × 12 × 12 × 12 = 2,985,984.

Let’s come back to the problem statement and understand how it will be solved, the problem statement asked to simplify 3 to the 6th power. It means the question asks to solve the power 6 of 3, which is represented as 36,

36 = 3 × 3 × 3 × 3 × 3 × 3

= 81 × 9

= 729

Therefore, 729 is the 6th power of 3.

### Sample Problem

Question 1: Solve the expression 43 – 23.

Solution:

To solve the expression, first, solve the 3rd powers on the numbers and then subtract the second term by the first term. However, the same problem can be solved in an easier way by simply applying a formula, the formula is,

x3 – y3 = (x – y)(x2 + y2 + xy)

43 – 23 = (4 – 2)(42 + 22 + 4 × 2)

= 2 × (16 + 4 + 8)

= 2 × 28

= 56

Question 2: Solve the expression 112 – 52.

Solution:

To solve the expression, first, solve the 2nd powers on the numbers and then subtract the second term by the first term. However, the same problem can be solved in an easier way by simply applying a formula, the formula is,

x2 – y2 = (x + y)(x – y)

112 – 52 = (11 + 5)(11 – 5)

= 16 × 6

= 96

Question 3: Solve the expression 33 + 93.

Solution:

To solve the expression, first, solve the 3rd powers on the numbers and then subtract the second term by the first term. However, the same problem can be solved in an easier way by simply applying a formula, the formula is,

x3 + y3 = (x + y)(x2 + y2 – xy)

33 + 93 = (9 + 3)(32 + 92 – 3 × 9)

= 12 × (9 + 81 – 27)

= 12 × 63

= 756

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