Evaluate (4/3)-3

In math, we use the terms “exponents and powers” when a number is multiplied by itself by a specific number of times. The number of times the number is multiplied by itself is equal to a number raised to the power of a natural number. For example, if a number “a” is multiplied by itself m times, then the expression obtained is defined as “a to the power of the m, or a raised to m”. Here, “a” is the base and “m” is the exponent. The base of an exponential expression refers to the number that is multiplied repeatedly by itself, while the exponent refers to the number of times the number is being multiplied. For example, 4 × 4 × 4 × 4 × 4 = 1024, where it can be written as 4⁵ in its exponential form. Here, 4⁵ means the number “4” is multiplied by itself by “5” times, “4” is the base number, and “5” is the exponent, and we read it as “4 raised to the power of 5”.

“a raised to the power of m”

a^m = a × a × a × a ×…….× a (m times)

“a” is the base of a^m

“m” is the exponent of a^m

Types of Exponents

There are various types of exponents depending upon the value of the power, they are:

• Positive exponent: A positive exponent can be simplified by multiplying the base by itself as many times as indicated by the exponent or power. For example, 3⁵ is equal to (3 × 3 × 3 × 3 × 3), i.e., 3 is multiplied by itself 5 times.
• Negative exponent: A negative exponent is similar to the positive power of the exponent, but the difference is that the value of the expression that has a negative exponent is the reciprocal of the value obtained in the case of a positive exponent. For example, the value of 3^-5 is equal to 1/3⁵.
• Zero exponents: The value of any expression whose exponent is 0 is equal to 1, and we do not need to consider the value of the base while simplifying the given exponential expression. For example, the values of (2/3)⁰, (-8)⁰, and (151)⁰ are 1.
• Rational exponent: Exponents that are rational or fractional will change into roots or radicals. For example, “5^1/2” can be written as “square root of 5”, and “2^1/3” is written as ” cube root of 2.

Laws of Exponents and Powers

Various laws of exponent and powers are used to simplify complex equations some of them are listed below:

Zero power rule	a0 = 1
Negative exponent rule	a-n = 1/an
Product Law of Exponents	am × an = a(m + n)
Quotient Rule of Exponents	am/an = a(m–n)
Power of a power rule	(am)n = a(m × n)
Power of a product rule	am × bm = (a × b)m
Power of a Quotient Rule	am/bm = (a/b)m
Fraction Exponent Rule	a1/n = n√a am/n = n√am

Evaluate (4/3)^-3

Solution:

Given expression (4/3)^-3

According to the negative law of exponents, we have

a^-n = 1/aⁿ

So,  (4/3)^-3 = 1/( (4/3)³

(4/3)³ =  (4/3) × (4/3) × (4/3)

= (4 × 4 × 4)/(3 × 3 × 3) = 64/27

So, 1/( (4/3)³ = 1/(64/27)

= 27/64

Hence,

the value of (4/3)^-3 = 27/64.

Solved Examples based on Exponent and Powers

Example 1: What is the exponent and power in the expression (x + y)³?

Solution:

The given expression is (x + y)³.

The base of the given exponential expression is “(x + y)”, while the exponent is “3”, i.e., the given expression is read as “(x + y) is raised to the power of 3.”

Example 2: Evaluate (12)^–8 × (12)¹⁰.

Solution:

Given: (12)^–8 × (12)¹⁰

We know that, a^m × aⁿ = a^{(m + n)}

So, (12)^–8× (12)¹⁰ = (12)^(-8+10)

= (12)² =12 × 12 = 144

Hence, (12)^–8 × (12)¹⁰ = 144.

Example 3: Find the value of x in the given expression: (7)¹²/(7)^-4 = (7)^3x–5.

Solution:

Given: (7)²⁰/(7)^-4 = (7)^3x–5

We know that, a^m/aⁿ = a^(m–n)

⇒ (7)^12–(–4) = (7)^3x–5

⇒(7)¹⁶ = (7)^3x–5

By comparing the exponents of the similar base, we get

⇒ 16 = 3x – 5

⇒ 3x = 16 + 5 = 21

⇒ x = 21/3 = 7

Thus, the value of x is 7.

Example 4: What does 2 to the power of 7 mean?

Solution:

Let us calculate the value of 2 to the power 7, i.e., 2⁷.

From the power rule of exponents, we have

aⁿ = a × a × a… n times

Hence, we can write 2⁷ as 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128

Thus, the value of 2 to the power of 7, i.e., 2⁷ is 128.

Example 5: What is the power notation of 16,807‬?

Solution:

The given number is 16,807‬.

On the prime factorization of 16,807, we get 14641 = 7 × 7 × 7 × 7 × 7,

i.e., 16,807 = 7 × 7 × 7 × 7 × 7  = 7⁵

Hence, 16,807‬ is equal to 7⁵ or we can write it as “7 raised to the power 5”.

FAQs on Exponents and Power

Questions 1:What are powers and exponents?

Answer:

An expression used to represent repeated multiplication is termed power and exponents. Generally, x^y is an expression where x is the base and y is the exponent.
For example, 2⁴ is the expression that shows that 2 is multiplied four times by itself.

Question 2: What are the examples of exponents?

Answer:

Some examples of exponents are:

3× 3 × 3 = 3³ (3 raised to 3rd power)
7 × 7 × 7 ×7 = 7⁴ (7 raised to 4th power)

Question 3: What are the rules of exponents?

Answer:

Some rules for simplifying exponents are:

a⁰ = 1
a^m × aⁿ = a^m+n
a^m/aⁿ = a^m-n
(a^m)ⁿ = a^mn

Question 4: What are negative exponents?

Answer:

Negative exponent is used to explain reciprocal of exponents. Suppose, 1/n is given it is expressed as n^-1, where (-1) is the exponent. If some number is raised to negative power then it represents the reciprocal of it. For example, 5 raised to -3 is represented by 5^-3, which is equal to 1/5³.

Question 6: What do we get if the exponent is 1 or 0?

Answer:

Suppose the exponent of a base is 1, then the value of the base remains unchanged. For example, 2¹ = 2
Suppose the exponent is 0, then we get answer as 1. For example, 7⁰ = 1