Find the value of x for 2x ÷ 2-4 = 45

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 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 = 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 × 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

BODMAS Rule

BODMAS rule is an acronym used for the rule of simplifying mathematics where B = Brackets, O = Order, D = Division, M = Multiplication, A = Addition, S = Subtraction. Therefore, BODMAS tells the order of simplification. If the calculations are not done in this order, the result can be wrong. Following the BODMAS rule, let’s solve the problem statement,

Find the value of x for which 2^x ÷ 2 ^-4 = 4⁵

Solution:

As it can be seen, on the right hand side, 4⁵ can be written as (2²)⁵

Therefore, 2^x ÷ 2^-4 = (2²)⁵

Now, using Power rule on left hand side,

2^x ÷ 2^-4 = 2¹⁰

2^x = 2¹⁰ × 2^-4

Now, using product rule on right hand side,

2^x = 2^{(10 – 4)}

2^x = 2⁶

The base values are same and hence, the exponents are equal on both sides too,

x = 6

Similar Problems

Question 1: Simplify x – 400 = 500 × 2.

Solution:

According to BODMAS rule, multiplication is done first,

x – 400 = 1000

Now, bring -400 on the left hand side,

x = 1400

Question 2: Simplify x + 7x – 7x – 49 = 10.

Solution:

According to BODMAS rule, do the addition first,

8x – 7x – 49 = 10

Now subtract 7x from 8x and bring -49 on right hand side,

x = 59

Question 3: Simplify x – 16 = 2 × 2.

Solution:

According to BODMAS rule, multiplication is done first,

x – 16 = 4

Now, bring -16 on the left hand side,

x = 20