1. Divide $\sqrt[3]{128}$ by $\sqrt[5]{64}$.
Without actually calculating, find the value of the following
(i) $96 \times 104$
(ii) $995 \times 995$
Factorise
(i) $x^3-2 x^2-x+2$
(ii) $x ^3+13 x ^2+32 x +20$
Locate $\sqrt{3}$ on the number line.
Answer (2)
We divided \(\frac{\sqrt[3]{128}}{\sqrt[5]{64}} = 2^{\frac{17}{15}}); found \(\text{96} \times \text{104} = 9984) and \(\text{995} \times \text{995} = 990025); factorized two polynomials, and located \(\text{√3}) using a right triangle.
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Divide radicals by expressing them as powers of 2 and simplifying the exponents: 5 64 3 128 = 2 15 17 .
Multiply without direct calculation using the difference of squares: 96 × 104 = ( 100 − 4 ) ( 100 + 4 ) = 9984 .
Factorise polynomials by grouping and identifying roots: x 3 − 2 x 2 − x + 2 = ( x − 1 ) ( x + 1 ) ( x − 2 ) and x 3 + 13 x 2 + 32 x + 20 = ( x + 1 ) ( x + 2 ) ( x + 10 ) .
Locate 3 on the number line by constructing a right-angled triangle with sides 1 and 2 , resulting in a hypotenuse of length 3 .
2 15 17 , 9984 , 990025 , ( x − 1 ) ( x + 1 ) ( x − 2 ) , ( x + 1 ) ( x + 2 ) ( x + 10 ) , locate 3 geometrically
Explanation
Problem Overview We are given four problems: dividing radicals, multiplying without direct calculation, factorizing polynomials and locating a square root on the number line. We will address each one step by step.
Dividing Radicals First, we need to divide 3 128 by 5 64 . We can express 128 and 64 as powers of 2: 128 = 2 7 and 64 = 2 6 . Therefore, we have 5 64 3 128 = 5 2 6 3 2 7 = 2 5 6 2 3 7 = 2 3 7 − 5 6 = 2 15 35 − 18 = 2 15 17 This can also be written as 15 2 17 = 2 15 2 2 = 2 15 4 . The approximate value is 2.1936.
Multiplication using Difference of Squares Next, we need to find the value of 96 × 104 without direct calculation. We can use the difference of squares formula: ( a − b ) ( a + b ) = a 2 − b 2 . We can express 96 and 104 as ( 100 − 4 ) and ( 100 + 4 ) respectively. Therefore, 96 × 104 = ( 100 − 4 ) ( 100 + 4 ) = 10 0 2 − 4 2 = 10000 − 16 = 9984
Squaring using Algebraic Identity Now, we need to find the value of 995 × 995 without direct calculation. We can rewrite it as ( 1000 − 5 ) 2 . Using the formula ( a − b ) 2 = a 2 − 2 ab + b 2 , we have 995 × 995 = ( 1000 − 5 ) 2 = 100 0 2 − 2 × 1000 × 5 + 5 2 = 1000000 − 10000 + 25 = 990025
Factorising by Grouping We need to factorise x 3 − 2 x 2 − x + 2 . We can factor by grouping: x 3 − 2 x 2 − x + 2 = x 2 ( x − 2 ) − 1 ( x − 2 ) = ( x 2 − 1 ) ( x − 2 ) = ( x − 1 ) ( x + 1 ) ( x − 2 )
Factorising Cubic Polynomial Next, we need to factorise x 3 + 13 x 2 + 32 x + 20 . We look for integer roots by testing factors of 20. Let's try x = − 1 :
( − 1 ) 3 + 13 ( − 1 ) 2 + 32 ( − 1 ) + 20 = − 1 + 13 − 32 + 20 = 0 So, x = − 1 is a root, and ( x + 1 ) is a factor. Now, we perform polynomial division to find the remaining quadratic factor: x + 1 x 3 + 13 x 2 + 32 x + 20 = x 2 + 12 x + 20 Now we factor the quadratic: x 2 + 12 x + 20 = ( x + 2 ) ( x + 10 ) Therefore, the complete factorization is: x 3 + 13 x 2 + 32 x + 20 = ( x + 1 ) ( x + 2 ) ( x + 10 )
Locating Square Root on Number Line Finally, we need to locate 3 on the number line. We know that 3 = 1 + 2 = 1 2 + ( 2 ) 2 . So, we construct a right-angled triangle with sides of length 1 and 2 . First, construct a right-angled triangle with sides of length 1 and 1. The hypotenuse will have length 1 2 + 1 2 = 2 . Then, using this 2 as one side and 1 as the other side, construct another right-angled triangle. The hypotenuse of this triangle will be ( 2 ) 2 + 1 2 = 2 + 1 = 3 . Use a compass to transfer this length to the number line, starting from 0.
Final Answers In summary:
5 64 3 128 = 2 15 17 = 2 15 4 ≈ 2.1936
96 × 104 = 9984
995 × 995 = 990025
x 3 − 2 x 2 − x + 2 = ( x − 1 ) ( x + 1 ) ( x − 2 )
x 3 + 13 x 2 + 32 x + 20 = ( x + 1 ) ( x + 2 ) ( x + 10 )
3 can be located on the number line by constructing a right-angled triangle with sides 1 and 2 , then transferring the length of the hypotenuse to the number line.
Examples
These mathematical concepts are useful in various real-life situations. For example, calculating with radicals is essential in physics and engineering when dealing with wave functions or signal processing. Factoring polynomials is used in cryptography and coding theory. The difference of squares simplifies calculations in financial analysis, such as calculating compound interest or discounts. Geometric constructions, like locating square roots on a number line, are fundamental in architecture and design for precise measurements and spatial arrangements. These skills collectively enhance problem-solving abilities across diverse fields, fostering innovation and efficiency.
