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equivalent expression

Equivalent expressions are like duplicates of each other but look different from the outside. The official statement state that “Equivalent expressions are expressions that work the same even though they look different.”

If you plugged any value in an equivalent expression, it will give the same result, it doesn’t matter how much different they look, they are the same expression, hence, it is named as equivalent expression.

There are most preferred two ways of solving the “Equivalent Expression” problem, either by putting the value of x into two given expressions and if they yield the same result then we got the correct equivalent expression for the given expression problems.

And another way to determine Equivalent Expressions is to manipulate the expressions into more simplified forms and then check to see if they are the same expression.

Let’s take a simple example before proceeding to the complex equivalent expressions problem.

⇒ Are the two expressions 10x + 5x +8 + 2 and 5(3x + 2) equivalent?

Given expression = 10x + 5x + 8 + 2 and 5(3x + 2)

Let’s try to put any value for x in the above expression.

Put x = 2

⇒ 10(2) + 5(2) + 8 + 2 = 40

⇒ 5(3×2 + 2) = 40

∴ So, both expressions are getting the same result for substituting the x variable as the value of 2.

Hence, the above expression is equivalent to each other and it doesn’t matter how much they look different.

Equivalent expression examples? 

Now we will try to solve 10 questions of equivalent expression that is from an easy to a moderate level which helps to grasp the concept and make you prepare for a harder one.

Q.1 Expression equivalent to 2x + 3y?

(a). 2y + 3x

(b).y + y + x + x + x

(c). y + y + y + x + x

(d). None of the above

Answer = (c) = y + y + y + x + x

Explanation:-

Give expression, 2x + 3y

Let’s put the value in given expression for x = 1 and for y = 2.

∴ 2(1) + 3(2) = 8

Now put the same value of x and y in given options

Option (a) = 2(2) + 3(1) = 7

Option (b) = 2 + 2 + 1 + 1 + 1 = 7

Option (c) = 2 + 2 + 2 + 1 + 1 = 8

Hence, the option (c) expression(y + y + y + x + x) equivalent with the given expression(2x + 3y)

Q.2 Expression equivalent to the square root of 80?

(a). 4√5

(b). 5√4

(c). 4√16 × √36

(d). Both (a) and (c)

Answer = (a) = 4√5

Explanation:-

Give expression, √80

Let, x = √80

Square both sides

∴ x2 = 80

Now square the each option

option (a) = 425 = 80

option (b) = 524 = 100

option (c) = 42 × 16 × 36 = 9,216

Hence, the option (a) is the right answer.

Q.3 Expression equivalent to 4b?

(a). b(3 + 1)

(b). b+2(b+2b)

(c). b + b2

(d). None of the above

Answer = (a) = b(3 + 1)

Explanation:-

Give expression, 4b

Let’s put the value in given expression for b = 2

∴ 4(2) = 8

Now put the same value of b in given options

Option (a) = 2(3 + 1) = 8

Option (b) = 2 + 2(2 + 2×2) = 14

Option (c) = 2 + 22 = 6

Hence, option (a) is correct for the given expression.

Q.4 Expression equivalent to (3x2)3?

(a). 32/x4

(b). 33/x4

(c). 27/x6

(d). None of the above

Answer = (c) = 27/x6

Explanation:-

Give expression, (3x2)3

We have inside the brackets “3 power 1” and “x power -2”

= (3x2)3

A value with a negative exponent is equivalent to its multiplicative inverse

= (3/x2)3

We use exponents laws: all exponents inside the brackets multiply with 3

= 33/x2 × 3

Finally, we have

= 27/x6

Hence, the expression (3x2)3 is equivalent to 27/x6

We can also solve this problem by putting the value for x.

Let’s put the value in given expression(3x2)3 for x = 2

∴ (3 × 2-2)3 = 27/64

Now put the same value of x in given options

Option (a) = 32/24 = 9/16

Option (b) = 33/24 = 27/16

Option (c) = 27/26 = 27/64

Hence, the option (c) is correct.

Q.5 Expression equivalent to 4(x + 1)?

(a). 4x + 2

(b). 2(2x + 2)

(c). 2(2x + 1)

(d). None of the above

Answer = (b) = 2(2x + 2)

Explanation:-

Give expression, 4(x + 1)

Let’s put the value in given expression for x = 1.

∴ 4(1 + 1) = 8

Now put the same value of x in given options

Option (a) = 4(1) + 2 = 6

Option (b) = 2(2×1 + 2) = 8

Option (c) = 2(2×1 + 1) = 6

Hence, the option (b) is correct.

Q.6 Expression equivalent to 15x + 25x2?

(a). 5x (3 + 5x)

(b). – 5(3 + 7x)

(c). – 7(5x2 + 2x)

(d). None of the above

Answer = (a) = 5x (3 + 5x)

Explanation:-

Give expression, 15x + 25x2

Let’s put the value in given expression for x = 1.

∴ 15(1) + 25(12) = 40

Now put the same value of x in given options

Option (a) = 5×1(3 + 5×1) = 40

Option (b) = -5(3 + 7×1) = -50

Option (c) = -7(5×12 + 2×1) = -49

Hence, option (a) is equivalent to the given expression.

Q.7 Expression equivalent to 3(-5y + 4)?

(a). 15y – 12

(b). 15 + 12y

(c). -15y + 12

(d). None of the above

Answer = (c) = -15y + 12

Explanation:-

Give expression, 3(-5y + 4)

To solve this, use distributive law.

⇒ A(B + C) = A.B + A.C

∴ 3(-5y + 4) = 3.(-5y) + 3.4

= -15y + 12

Hence, option (C) is correct for the given expression.

Q.8 Expression equivalent to (4 + 7i)(3 + 4i)?

(a). 16 – 37i

(b). -16 + 37i

(c). 12 – 28i

(d). None of the above

Answer = (b) = -16 + 37i

Explanation:-

Give expression, (4 + 7i)(3 + 4i)

As we know, i2 = -1

Use the distributive property to solve this expression-

⇒ A(B + C) = A.B + A.C

∴ (4 + 7i)(3 + 4i) = 4*3 + 4*4i + 7i*3 + 7i*4i

= 12 + 16i + 21i + 28i2

= 12 + 37i + 28*(-1)

= 12 – 28 + 37i

= -16 + 37i

Hence, option (b) is a correct equivalent expression for the given expression.

Q.9 Expression equivalent to log18 – log(p + 2)?

(a). log [18/(p + 2)]

(b). log18.log(p+2)

(c). log(18 – p – 2)

(d). None of the above

Answer = (a) = log [18/(p + 2)]

Explanation:-

Give expression, log18 – log(p + 2)

In logarithm if the expression is in the form of log(a/b), then we can write the expression in the form like log(a/b) = loga – logb 
Using the property of logarithmic, we will solve the above expression.
∴ log18 – log(p + 2)

= log [18/(p + 2)]           [∴ loga – logb  = log(a/b)]

Hence, the option(a) is correct for above expression.

Q.10 Expression equivalent to (f + g)(4)?

(a). f(x) + g(4)

(b). f(4) + g(4)

(c). 4(f.g)

(d). None of the above

Answer = (b) = f(4) + g(4)

Explanation:-

Give expression, (f + g)(4)

Use the distributive property to solve this expression-

⇒ A(B + C) = A.B + A.C

∴ (f + g)(4) =  f(4) + g(4)

Hence, the expression (f + g)(4) is equal to f(4) + g(4).

Equivalent expression examples related to complex fraction

“Complex fractions are fractions with a numerator, denominator, or both that are also fractions”

Here, we will solve the 3 problems that will help you in grasping the concept of “Equivalent expression complex fraction” problems.

Q.1 Expression equivalent to x/3/(x – 3)?

(a). x/3(x – 9)

(b). x/(3x – 9)

(c). (3x2 – 9)

(d). None of the above

Answer = (b). x/(3x – 9)

Explanation:-

Give expression, x/3/(x – 3)

Let’s put the value in given expression for x = 2.

∴ 2/3/(2 – 3) = -2/3

Now put the same value of x in given options

Option (a) = 2/3(2 – 9) = -2/21

Option (b) = 2(3×2 -9) = -2/3

Option (c) = (3×22 – 9) = 3

Hence, option (b) is equivalent to the given complex expression.

Q.2 Expression equivalent to (1÷x5/1÷x6)4?

(a). 4x

(b). x4

(c). x-4

(d). None of the above

Answer = (b) = x4

Explanation:-

Give expression, (1÷x5/1÷x6)4

∴ (x)4              (∴A÷B/C÷D = A÷B × D÷C)

= x4

Hence, option (b) is correct for the given expression.

Q.3 Which expression is equivalent to which expression is equivalent to the complex fraction

(a). 1 – a

(b). -1

(c). +1

(d). 1 + a

Answer = (a) = 1 – a

Explanation:-

Give expression, (1/a – a)/(1/a + 1)

∴ (1 – a²)/(1 + a)

= (1 – a)(1 + a)/(1+a)            [∴ a2 – b2 = (1 + a)(1 + b)]

= (1 – a)

Hence, the correct option is (a) for the given expression.

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