# What is 3 to the 4th power?

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 7^{5}. Here, 7 is the base value and 5 is the exponent and the value is 16807. 11 × 11 × 11, can be written as 11^{3}, here, 11 is the base value and 3 is the exponent or power of 11. The value of 11^{3} 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 cx^{y} 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 = p ^{n}**

**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 ⇢ a
^{n}+ a^{m}= a^{n + m} - Quotient Rule ⇢ a
^{n}/ a^{m}= a^{n – m} - Power Rule ⇢ (a
^{n})^{m}= a^{n × m}or^{m}√a^{n}= a^{n/m} - Negative Exponent Rule ⇢ a
^{-m }= 1/a^{m} - Zero Rule ⇢ a
^{0}= 1 - One Rule ⇢ a
^{1}= a

### What is 3 to the 4^{th} power?

**Solution**:

Any number having a power of 4 can be written as the quartic of that number. The quartic of a number is the number multiplied by itself four times, quartic of the number is represented as the exponent 4 on that number. If quartic of x has to be written, it will be x

^{4}. For instance, the quartic of 5 is represented as 5^{4}and is equal to 5 × 5 × 5 × 5 = 625. Another example can be the quartic of 12, represented as 12^{4}, is equal to 12 × 12 × 12 × 12 = 20736.Let’s come back to the problem statement and understand how it will be solved, the problem statement asked to simplify 3 to the 4

^{th}power. It means the question asks to solve the quartic of 3, which is represented as 3^{4},3

^{4}= 3 × 3 × 3 × 3= 9 × 3 × 3

= 81

Therefore, 81 is the 4

^{th}power of 3.

### Sample Problem

**Question 1: Solve the expression 6 ^{3} – 2^{3}.**

**Solution:**

To solve the expression, first solve the 3

^{rd}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,x

^{3}– y^{3}= (x – y)(x^{2}+ y^{2}+ xy)6

^{3}– 2^{3}= (6 – 2)(6^{2}+ 2^{2 }+ 6 × 2)= 4 × (36 + 4 + 12)

= 4 × 52

= 208

**Question 2: Solve the expression 7 ^{2} – 5^{2}.**

**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,

x

^{2}– y^{2}= (x + y)(x – y)7

^{2}– 5^{2}= (7 + 5)(7 – 5)= 12 × 2

= 24

**Question 3: Solve the expression 3 ^{3 }+ 3^{3}.**

**Solution:**

To solve the expression, first solve the 3

^{rd}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,x

^{3}+ y^{3}= (x + y)(x^{2}+ y^{2}– xy)3

^{3}+ 3^{3 }= (3 + 3)(3^{2}+ 3^{2}– 3 × 3)= 6 × (9 + 9 – 9)

= 6 × 9

= 54

Another method of solving it is to simply calculate the cube of each term and then add both the terms,

3

^{3}+ 3^{3}= 27 + 27= 54