Find the values using suitable identities: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

Understand the Question To evaluate products and powers of numbers without direct multiplication, express the numbers relative to a convenient base (such as ) and apply standard algebraic identities: Product of Binomials: Difference of Squares: Cube of a Sum: Cube of a Difference: Negative Base Property: Part (i) Evaluate Step 1 · Apply the Identity Write as . Using the identity , where , , and : Answer > (i) Part (ii) Evaluate Step 1 · Apply the Difference of Squares Identity Write as . Using the identity , where and : Answer > (ii) Part (iii) Evaluate Step 1 · Apply the Difference of Squares Identity Write as . Using the identity , where and : Answer > (iii) Part (iv) Evaluate Step 1 · Apply the Cube of a Difference Identity Write as . Using the identity , where and : Answer > (iv) Part (v) Evaluate Step 1 · Apply the Cube of a Difference Identity Write as . Using the identity , where and : Answer > (v) Part (vi) Evaluate Step 1 · Apply the Cube of a Difference Identity Write as . Using the identity , where and : Answer > (vi) Part (vii) Evaluate Step 1 · Apply the Cube of a Sum Identity Using the property : Using the identity , where and : Therefore: Answer > (vii) Part (viii) Evaluate Step 1 · Apply the Cube of a Difference Identity Using the property : Using the identity , where and : Therefore: Answer > (viii) Common Mistakes Signs in expansion: Mixing up the alternating signs. Remember that the middle terms alternate: . Cubing Negative Numbers: Forgetting that an odd power of a negative number remains negative, i.e., .

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Question 2

Find the values using suitable identities:

(i) $17 \times 21$

(ii) $104 \times 96$

(iii) $24 \times 16$

(iv) $147^3$

(v) $199^3$

(vi) $127^3$

(vii) $(-107)^3$

(viii) $(-299)^3$

Solution

Understand the Question

To evaluate products and powers of numbers without direct multiplication, express the numbers relative to a convenient base (such as $10, 20, 100, 200, 300$ ) and apply standard algebraic identities:

Product of Binomials: $(x + a)(x + b) = x^2 + (a + b)x + ab$
Difference of Squares: $(a + b)(a - b) = a^2 - b^2$
Cube of a Sum: $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$
Cube of a Difference: $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$
Negative Base Property: $(-x)^3 = -(x^3)$

(i) Evaluate $17 \times 21$

Step 1 · Apply the Identity

Write $17 \times 21$ as $(20 - 3)(20 + 1)$ .

Using the identity $(x + a)(x + b) = x^2 + (a + b)x + ab$ , where $x = 20$ , $a = -3$ , and $b = 1$ :

\begin{aligned} 17 \times 21 &= (20 - 3)(20 + 1) \\[0.6em] &= 20^2 + (-3 + 1)(20) + (-3)(1) \\[0.6em] &= 400 + (-2)(20) - 3 \\[0.6em] &= 400 - 40 - 3 \\[0.6em] &= 357 \end{aligned}

Answer

(i) $357$

(ii) Evaluate $104 \times 96$

Step 1 · Apply the Difference of Squares Identity

Write $104 \times 96$ as $(100 + 4)(100 - 4)$ .

Using the identity $(a + b)(a - b) = a^2 - b^2$ , where $a = 100$ and $b = 4$ :

\begin{aligned} 104 \times 96 &= (100 + 4)(100 - 4) \\[0.6em] &= 100^2 - 4^2 \\[0.6em] &= 10000 - 16 \\[0.6em] &= 9984 \end{aligned}

Answer

(ii) $9984$

(iii) Evaluate $24 \times 16$

Step 1 · Apply the Difference of Squares Identity

Write $24 \times 16$ as $(20 + 4)(20 - 4)$ .

Using the identity $(a + b)(a - b) = a^2 - b^2$ , where $a = 20$ and $b = 4$ :

\begin{aligned} 24 \times 16 &= (20 + 4)(20 - 4) \\[0.6em] &= 20^2 - 4^2 \\[0.6em] &= 400 - 16 \\[0.6em] &= 384 \end{aligned}

Answer

(iii) $384$

(iv) Evaluate $147^3$

Step 1 · Apply the Cube of a Difference Identity

Write $147$ as $(150 - 3)$ .

Using the identity $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$ , where $a = 150$ and $b = 3$ :

\begin{aligned} 147^3 &= (150 - 3)^3 \\[0.6em] &= 150^3 - 3(150)^2(3) + 3(150)(3)^2 - 3^3 \\[0.6em] &= 3375000 - 9(22500) + 450(9) - 27 \\[0.6em] &= 3375000 - 202500 + 4050 - 27 \\[0.6em] &= 3176523 \end{aligned}

Answer

(iv) $3176523$

(v) Evaluate $199^3$

Step 1 · Apply the Cube of a Difference Identity

Write $199$ as $(200 - 1)$ .

Using the identity $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$ , where $a = 200$ and $b = 1$ :

\begin{aligned} 199^3 &= (200 - 1)^3 \\[0.6em] &= 200^3 - 3(200)^2(1) + 3(200)(1)^2 - 1^3 \\[0.6em] &= 8000000 - 3(40000) + 600 - 1 \\[0.6em] &= 8000000 - 120000 + 600 - 1 \\[0.6em] &= 7880599 \end{aligned}

Answer

(v) $7880599$

(vi) Evaluate $127^3$

Step 1 · Apply the Cube of a Difference Identity

Write $127$ as $(130 - 3)$ .

Using the identity $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$ , where $a = 130$ and $b = 3$ :

\begin{aligned} 127^3 &= (130 - 3)^3 \\[0.6em] &= 130^3 - 3(130)^2(3) + 3(130)(3)^2 - 3^3 \\[0.6em] &= 2197000 - 9(16900) + 390(9) - 27 \\[0.6em] &= 2197000 - 152100 + 3510 - 27 \\[0.6em] &= 2048383 \end{aligned}

Answer

(vi) $2048383$

(vii) Evaluate $(-107)^3$

Step 1 · Apply the Cube of a Sum Identity

Using the property $(-x)^3 = -(x^3)$ :

$(-107)^3 = -(107^3) = -(100 + 7)^3$

Using the identity $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$ , where $a = 100$ and $b = 7$ :

\begin{aligned} (100 + 7)^3 &= 100^3 + 3(100)^2(7) + 3(100)(7)^2 + 7^3 \\[0.6em] &= 1000000 + 21(10000) + 300(49) + 343 \\[0.6em] &= 1000000 + 210000 + 14700 + 343 \\[0.6em] &= 1225043 \end{aligned}

Therefore:

$(-107)^3 = -1225043$

Answer

(vii) $-1225043$

(viii) Evaluate $(-299)^3$

Step 1 · Apply the Cube of a Difference Identity

Using the property $(-x)^3 = -(x^3)$ :

$(-299)^3 = -(299^3) = -(300 - 1)^3$

Using the identity $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$ , where $a = 300$ and $b = 1$ :

\begin{aligned} (300 - 1)^3 &= 300^3 - 3(300)^2(1) + 3(300)(1)^2 - 1^3 \\[0.6em] &= 27000000 - 3(90000) + 900 - 1 \\[0.6em] &= 27000000 - 270000 + 900 - 1 \\[0.6em] &= 26730899 \end{aligned}

Therefore:

$(-299)^3 = -26730899$

Answer

(viii) $-26730899$

Common Mistakes

Signs in $(a - b)^3$ expansion: Mixing up the alternating signs. Remember that the middle terms alternate: $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$ .
Cubing Negative Numbers: Forgetting that an odd power of a negative number remains negative, i.e., $(-x)^3 = -x^3$ .

Question 2

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