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Problem Set 2 (Surds and Indices) - Quantitative Aptitude Video Lecture
FAQs on Problem Set 2 (Surds and Indices) - Quantitative Aptitude Video Lecture - CSAT Preparation
| 1. What are surds and how are they used in mathematics? |  |
Ans. Surds are irrational numbers that cannot be expressed as a simple fraction or a decimal. They are often represented by the symbol √. In mathematics, surds are used to represent the square roots of non-perfect square numbers or any other roots of non-perfect nth power numbers.
| 2. How do we simplify surds? | |
Ans. To simplify surds, we follow certain rules. For example, if we have a surd like √12, we can simplify it by finding the largest perfect square that divides 12 evenly. In this case, it is √4, which simplifies to 2. So, √12 can be simplified to 2√3.
| 3. What are indices and how are they used in mathematics? | |
Ans. Indices, also known as exponents or powers, are used to represent repeated multiplication of the same number. They are written as a small number raised to the right and above the base number. For example, in 2³, the base is 2 and the index is 3. It means 2 multiplied by itself three times (2 x 2 x 2 = 8).
| 4. How do we simplify expressions with indices? | |
Ans. To simplify expressions with indices, we use certain rules. For example, when multiplying two numbers with the same base but different indices, we can add the indices. So, 2² x 2³ can be simplified to 2^(2+3) = 2^5 = 32. Similarly, when dividing two numbers with the same base but different indices, we can subtract the indices.
| 5. How are surds and indices used together in mathematics? | |
Ans. Surds and indices are often used together to solve complex mathematical problems. For example, when simplifying expressions with surds involving indices, we can apply the rules of indices to simplify the expression further. Similarly, when finding the square root or any other root of an expression with indices, we can apply the rules of surds to simplify the answer.
Text Transcript from Video
let us solve some more questions on
routes and indices what could be the
answer of this first question well you
can use the formula a raise to M times a
raise to n is equal to e raise to M plus
n so what we can do here is we can write
this as 256 raised to 0.16 plus 0.09 so
this is equal to 256 raised to 0.25 and
that can be written as 256 raised to 25
upon 125 upon 100 is 1 by 4 so this is
equivalent to 256 raised to 1 by 4
now you can reduce this further 256 is
equivalent to 4th power of 4 that is 4
raised to 4 is equal to 256 you can
calculate that this is equals to 4 raise
to 4 raised to 1 by 4 now you have
another formula a raise to M raise to n
it is equals to a raise to M times n so
this is equivalent to 4 raise to 4 times
1 by 4 so 4 & 4 gets cancelled out what
remains is 4 raised to 1 and that is 4
so the answer is option a moving on to
the second question what could be the
answer of this question okay let us
write this in simple terms we can write
0.04 as 4 upon 100 raise to minus 1.5 so
if you have a number raised to minus 1
or a negative power it is equals to 1
upon a raised to that power so this can
be written as you just have to invert
the numerators and denominators so this
is 100 upon 4 raised to 1.5 now 4 times
25 is 100 so this is
25 raise to 1.5 now 1.5 can be written
as we have 25 raise to teen by 10 so 5
times 3 is 15 5 times 2 is 10 so this is
equals to 25 is 5 square so phi square
raise to 3 by 2 and that is 5 raise to 2
times 3 by 2 another law of indices that
is used here so 2 + 2 gets cancelled out
what remains is 5 cube + 5 cubed is 125
so the answer is option B let us move on
to the next question what could be the
answer well you have a raise to M times
a raise to n is equals to a raise to M
plus n so you can use this here we can
write let us term this as X the question
mark so we can write this the left hand
side as 17 raised to 3.5 Plus this X is
equals to 17 raised to 8 so the numbers
are same we can write 3 point 5 plus X
is equals to 8 therefore X is equals to
8 minus 3.5 and that is equal to 4.5 so
we have the answer as 4.5 that was very
easy let's move on to the next question
what could be the answer of this this
one's also very easy you can write 36 as
6 square so you can write 6 square times
6 square times 6 square times 6 square
is equals to 6 raised to let's term this
as X now you know the formula a raise to
M times a raise to n is equals to a
raise to M plus n so this can be written
as 6 raised to 2 plus 2 plus 2 plus 2 is
equals to 6 raised to X so this is 2
plus 2 is 4 plus 2 is 6
plus 2 is 8 therefore X is equal to 8 so
we have the answer 8 now the fifth and
final question let's solve this and find
out the answer so you have 8 raise to
minus 25 minus 8 raised to minus 26 so
you know that L is 2 minus 1 is equal to
1 upon a that it's a raise to any
negative power is is the reciprocal of
that number raised to that power so we
can write this as 1 upon 8 raised to 25
minus 1 upon 8 raised to 26 so this is
equivalent to 8 raised to 20 6-8 raised
to 25 up on 8 raised to 25 times 8
raised to 26 now we can solve the
denominator but let's keep it as it is
you can take common 8 raised to 25 from
the numerator so if I take 8 raised to
25 common so I can write 8 raised to 25
times 8 is equals to 8 raised to 26 it
raised to 25 times it is it is 226 isn't
it - 1 8 raised to 25 times 1 is 8
raised to 25/8 raised to 25 times 8
raised to 26 now we can cancel out 8
raised to 25 and it is 225 so what
remains is 8 minus 1 that is 7 upon 8
raised to 26 and this is equivalent to 7
times 8 raised to minus 26 and that's
our answer right there option B
routes and indices what could be the
answer of this first question well you
can use the formula a raise to M times a
raise to n is equal to e raise to M plus
n so what we can do here is we can write
this as 256 raised to 0.16 plus 0.09 so
this is equal to 256 raised to 0.25 and
that can be written as 256 raised to 25
upon 125 upon 100 is 1 by 4 so this is
equivalent to 256 raised to 1 by 4
now you can reduce this further 256 is
equivalent to 4th power of 4 that is 4
raised to 4 is equal to 256 you can
calculate that this is equals to 4 raise
to 4 raised to 1 by 4 now you have
another formula a raise to M raise to n
it is equals to a raise to M times n so
this is equivalent to 4 raise to 4 times
1 by 4 so 4 & 4 gets cancelled out what
remains is 4 raised to 1 and that is 4
so the answer is option a moving on to
the second question what could be the
answer of this question okay let us
write this in simple terms we can write
0.04 as 4 upon 100 raise to minus 1.5 so
if you have a number raised to minus 1
or a negative power it is equals to 1
upon a raised to that power so this can
be written as you just have to invert
the numerators and denominators so this
is 100 upon 4 raised to 1.5 now 4 times
25 is 100 so this is
25 raise to 1.5 now 1.5 can be written
as we have 25 raise to teen by 10 so 5
times 3 is 15 5 times 2 is 10 so this is
equals to 25 is 5 square so phi square
raise to 3 by 2 and that is 5 raise to 2
times 3 by 2 another law of indices that
is used here so 2 + 2 gets cancelled out
what remains is 5 cube + 5 cubed is 125
so the answer is option B let us move on
to the next question what could be the
answer well you have a raise to M times
a raise to n is equals to a raise to M
plus n so you can use this here we can
write let us term this as X the question
mark so we can write this the left hand
side as 17 raised to 3.5 Plus this X is
equals to 17 raised to 8 so the numbers
are same we can write 3 point 5 plus X
is equals to 8 therefore X is equals to
8 minus 3.5 and that is equal to 4.5 so
we have the answer as 4.5 that was very
easy let's move on to the next question
what could be the answer of this this
one's also very easy you can write 36 as
6 square so you can write 6 square times
6 square times 6 square times 6 square
is equals to 6 raised to let's term this
as X now you know the formula a raise to
M times a raise to n is equals to a
raise to M plus n so this can be written
as 6 raised to 2 plus 2 plus 2 plus 2 is
equals to 6 raised to X so this is 2
plus 2 is 4 plus 2 is 6
plus 2 is 8 therefore X is equal to 8 so
we have the answer 8 now the fifth and
final question let's solve this and find
out the answer so you have 8 raise to
minus 25 minus 8 raised to minus 26 so
you know that L is 2 minus 1 is equal to
1 upon a that it's a raise to any
negative power is is the reciprocal of
that number raised to that power so we
can write this as 1 upon 8 raised to 25
minus 1 upon 8 raised to 26 so this is
equivalent to 8 raised to 20 6-8 raised
to 25 up on 8 raised to 25 times 8
raised to 26 now we can solve the
denominator but let's keep it as it is
you can take common 8 raised to 25 from
the numerator so if I take 8 raised to
25 common so I can write 8 raised to 25
times 8 is equals to 8 raised to 26 it
raised to 25 times it is it is 226 isn't
it - 1 8 raised to 25 times 1 is 8
raised to 25/8 raised to 25 times 8
raised to 26 now we can cancel out 8
raised to 25 and it is 225 so what
remains is 8 minus 1 that is 7 upon 8
raised to 26 and this is equivalent to 7
times 8 raised to minus 26 and that's
our answer right there option B
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