HOW TO SIMPLIFY EXPONENTIAL EXPRESSIONS

Problem 1 :

x⁴/x

Solution :

Step 1 :

x⁴/x¹

Step 2 :

= x⁴ · x^-1

Step 3 :

Since the bases are the same, use one base and combine the powers.

= (x^{4 – 1})

= x³

Problem 2 :

4b² × 2b³

Solution :

Step 1 :

4b² × 2b³

Step 2 :

Since the bases are the same, use one base and combine the powers and multiply the coefficients.

= 8(b^{2 + 3})

 = 8b⁵

Problem 3 :

a⁶b³/ a⁴b

Solution :

Step 1 :

a⁶b³/a⁴b¹

Step 2 :

= a⁶b³ · a^-4b^-1

Step 3 :

Since the bases are same, combine the powers.

= (a^{6 – 4} · b^{3 – 1})

= a² · b²

Problem 4 :

18x⁶/3x³

Solution :

Step 1 :

18x⁶/3x³

Step 2 :

Dividing the coefficients.

6x⁶/x³

Step 3 :

= 6(x⁶ · x^-3)

Since the bases are same, use one base and combine the powers.

= 6(x^{6 – 3})

= 6x³

Problem 5 :

5x³y²/15xy

Solution :

Step 1 :

5x³y²/15xy

Step 2 :

Dividing the coefficients.

= x³y²/3xy

Step 3 :

= 1/3 (x³y² · x^-1y^-1)

Since the bases are same, use one base and combine the powers.

= 1/3(x^{3 – 1}  · y^{2 – 1})

= 1/3 (x² · y)

Problem 6 :

24t⁶ r⁴/15t⁶r²

Solution :

24t⁶ r⁴/15t⁶r²

Dividing the coefficients.

= 8t⁶ r⁴/5t⁶r²

= 8/5(t⁶ r⁴ /t⁶r²)

= 8/5(t^{6 – 6} · r^{4 – 2})

= 8/5(t⁰ · r²)

= 8r²/5

Problem 7 :

3pq³ × 5p⁵

Solution :

= 3pq³ × 5p⁵

= 15(p^{1 + 5}  · q³)

= 15p⁶q³

Problem 8 :

x¹²/(x³)²

Solution :

x¹²/(x³)²

= x¹²/x⁶

= x¹² · x^-6

= x^{(12 – 6)}

= x⁶

Problem 9 :

(t⁶ × t⁴)/(t²)³

Solution :

(t⁶ × t⁴)/(t²)³

= (t⁶ × t⁴)/(t⁶)

= t⁴

Problem 10 :

If (-a² b³) (2ab²) (-3b) = ka^m bⁿ, what is the value of m + n ?

Solution :

(-a² b³) (2ab²) (-3b) = ka^m bⁿ

6a² ab²b³ b = ka^m bⁿ

Using the rules of exponents, simplifying it

6a²⁺¹ b²⁺³⁺¹ = ka^m bⁿ

6a³ b⁶ = ka^m bⁿ

Comparing the corresponding terms, we get

k = 6, m = 3 and n = 6

m + n = 3 + 6

= 9

So, the value of m + n is 9.

Problem 11 :

If (2/3 a²b)² (4/3 ab)^-3 = ka^m bⁿ, what is the value of k ?

Solution :

(2/3 a²b)² (4/3 ab)^-3 = ka^m bⁿ

Distributing the power, we get

(2/3)² (a²)²(b)² (4/3 ab)^-3 = ka^m bⁿ

(4/9) (a⁴)b² (4/3 ab)^-3 = ka^m bⁿ

To convert the negative exponent as positive exponent, we have to take the reciprocal.

(4/9) (a⁴)b² (3/4 ab)³ = ka^m bⁿ

(4/9) (a⁴)b² (3/4)³ a³b³ = ka^m bⁿ

(4/9) (27/64) a⁴b² a³b³ = ka^m bⁿ

(4/9) (27/64) a⁴⁺³b²⁺³ = ka^m bⁿ

(3/16) a⁷b⁵ = ka^m bⁿ

Comparing the corresponding terms, we get

k = 3/16, m = 7 and n = 5

Problem 12 :

If [x³ (-y)² z^-2]/x^-2y³z = x^m/yⁿz^p, what is the value of m + n + p?

Solution :

[x³ (-y)² z^-2]/x^-2y³z = x^m/yⁿz^p

[x³ y²/ z²] / [(1/x²)y³z] = x^m/yⁿz^p

[x³ x²y²/ z²] / [y³z] = x^m/yⁿz^p

[x³⁺² y²/ z²] / [y³z] = x^m/yⁿz^p

[x⁵ y²/ z²] / [y³z] = x^m/yⁿz^p

[x⁵ y²/ z²] ⋅ [1/y³z] = x^m/yⁿz^p

x⁵ ⋅ [1/y^3-2z²⁺¹] = x^m/yⁿz^p

x⁵/y¹z³ = x^m/yⁿz^p

Comparing the corresponding terms,

m = 5, n = 1 and p = 3

Problem 13 :

If 2^x = 5, what is the valu of 2^x + 2^2x + 2^3x ?

Solution :

= 2^x + 2^2x + 2^3x

= 2^x + (2^x)² + (2^x)³

Applying the value of 2^x = 5

= 5 + 5² + 5³

= 5 + 25 + 125

= 155

Problem 14 :

(6xy²)(2xy)² / (8x²y²)

If the expression above is written in the form ax^myⁿ, what is the value of a + m + n? 

Solution :

= (6xy²)(2xy)² / (8x²y²)

= (6xy²)(4x²y²) / (8x²y²)

= (24 x²⁺¹ y²⁺²) / (8x²y²)

= (24 x³ y⁴) / (8x²y²)

= 3x^3-2 y^4-2

ax^myⁿ = 3x¹ y²

Comparing the corresponding terms, we get

a = 3, m = 1 and n = 2

a + m + n = 3 + 1 + 2

= 6

So, the value of a + m + n is 6.

Problem 15 :

If 8,200 × 300,000  is equal to 2.46 x 10ⁿ, what is the value of n?

Solution :

= 8,200 × 300,000

= 8.2 x 10³ x 3 x 10⁵

= 24.6 x 10³⁺⁵

= 24.6 x 10⁸

= 2.46 10^-1 x 10⁸

= 2.46 x 10^-1+8

= 2.46 x 10⁷

So, the value of n is 7.

Problem 16 :

(3^x + 3^x + 3^x) 3^x Which of the following is equivalent to the expression shown above?

Solution :

= (3^x + 3^x + 3^x) 3^x

Adding the same quantities,

= (3 ⋅ 3^x) 3^x

= (3^x+1) 3^x

= 3^x+1+x

= 3^{2x + 1}

Problem 17 :

a + b = 4 and a - b = 2, if x > 1 and x^a²/x^b² = x^c, what is the value of c ?

Solution :

x^a²/x^b² = x^c

Given that a + b = 4 and a - b = 2

x^(a² -b²) = x^c

x^(a + b)(a - b) = x^c

x^(4)(2) = x^c

x⁸ = x^c

c = 8

So, the vlaue of c is 8.

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