Is 2³³ = 213³?

Exponents and powers are used to show very large numbers or very small numbers in a simplified manner. For example, if we have to show 2 × 2 × 2 × 2 in a simple way, then we can write it as 2⁴, where 2 is the base and 4 is the exponent. The whole expression 2⁴ is said to be power.

Exponents and Powers

Power is a value or an expression that represents repeated multiplication of the same number or factor. The number of times the base is multiplied by itself is the value of the exponent. For example,

• 3² = 3 raised to power 2 = 3 × 3 = 9
• 4³ = 4 raised to power 3 = 4 × 4 × 4 = 64

An exponent of a number represents the number of times the number is multiplied by itself. For example, 2 is multiplied by itself for n times,

2 × 2 × 2 × 2 × …n times = 2ⁿ

The above expression, 2ⁿ, is said as 2 raised to the power n. Therefore, exponents are also called power or sometimes indices.

General Form of Exponents

Exponent represents that how many times a number should be multiplied by itself to get the result. Thus any number ‘b’ raised to power ‘p’ can be expressed as:

b^p =  {b × b × b × b × …  × b} p times

Here b is any number and p is a natural number.

• b^p is also called the p^th power of b.
• ‘b’ is the base and ‘p’ is the exponent or index or power.
• ‘b’ is multiplied ‘p’ times, and thereby exponentiation is the shorthand method of repeated multiplication.

Laws of Exponents

Let ‘b’ is any number or integer (positive or negative) and ‘p₁’,  ‘p₂’ are positive integers, denoting the power to the bases.

• Multiplication Law: It states that the product of two exponents with the same base and different powers equals to base raised to the sum of the two powers or integers.

b^p₁ × b^p₂ = b^(p₁ ^{+ p}₂⁾

• Division Law: It states that if two exponents having the same bases and different powers are divided, then the results will be base raised to the difference between both powers.

b^p₁ ÷ b^p₂ = b^p₁/ b^p₂ = b^(p₁ ^{– p}₂⁾

• Negative Exponent Law: If the base has a negative power, then it can be converted into its reciprocal but with positive power or integer to the base.

b^-p = 1/b^p

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

Is 2^{33) = (213)³? 

Solution: 

LHS = 2³³, RHS = 213³

Let’s solve both sides,

Actually, it’s not required to solve any of the sides, as RHS is 213 i.e. odd, and powering it odd times will lead to an odd value. for example 3³ = 27 i.e. odd

And LHS has a base 2 and any power of 2 will always be an even value for example 2³ = 8, 2⁵ = 32.

So, if LHS is an even value and RHS is an odd value, both can’t be equal.

So, 2³³ is not equal to 213³. It can also be written as 2^{33) ≠ (213)³

Similar Problems 

Question 1: Simplify (-4a²/b³)³

Solution:

Here one can write above equation as,

= {(-4a²)³/ ( b³)³}

Now,

= {(-4)³ × (a²)³} / (b³)³

= {(-64) × (a^{2 × 3})} / b^{3 × 3} {Power Rule ⇢ (aⁿ)^m = a^{n × m}}

= -64a⁶/b⁹

Question 2: What is the product of (7x²y³) and (3x⁵y⁸)?

Solution:

The product of  (7x²y³) and (3x⁵y⁸)

 = (7x²y³) × (3x⁵y⁸)

 = (7x²y³) × (3x⁵y⁸)

 = 21 x²x⁵ × y³y⁸

 = 21x^{2 + 5} × y^{3 + 8} {Product Rule ⇢ aⁿ × a^m = a^{n + m}}

 = 21x⁷y¹¹

Question 3: What is x⁹ divided by x³?

Solution:

Here given x⁹ divided by x³

And use {Quotient Rule ⇢ aⁿ / a^m = a^{n – m}}

So write it as x⁹ / x³

= x^{9 – 3}

= x⁶

Question 4: Solve (2²) × (3²)

Solution:

Here when bases are different and powers are same. So as per the product rule we can write as aⁿ × bⁿ = (a × b)ⁿ

2² × 3²

= (2 × 3)²

= 6²

= 36