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Are there two irrational number whose sum and product both are rationals ? Justify
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Yes, and are two irrational numbers
Sum
Product
Prove that the negative of an irrational number is an irrational number.
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Let negative of an irrational number be rational number
Let P be an irrational number by the assumption P is a rational number. Since negative of a rational number is rational (P) = P is a rational number but we assumed P as an irrational number. So our assumption is wrong, so negative of an irrational number is irrational.
Find 4 rational numbers between \(\dfrac{1}{3} \) and \(\dfrac{4}{5} \).
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are 4 rational numbers
Express \(1.\bar{323} \) in the form \(\dfrac{p}{q} \), where \(p \) and \(q \) are integers \(q\ne0 \).
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Let
Simplify : \(\left(\dfrac{81}{16}\right)^{\frac{3}{4}}\times\left\{\left(\dfrac{25}{9}\right)^{\frac{3}{2}}\div\left(\dfrac{5}{2}\right)^{3}\right\} \)
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Show that \(\dfrac{x^{a(bc)}}{x^{b(ac)}}\div\left(\dfrac{x^b}{x^a}\right)^c=1 \).
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Locate \(\sqrt{13} \) on the number line.
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Represent \(\sqrt{5.6} \) on the number line.
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Presentation of on number line:
Mark the distance 5.6 units from a fixed points A on a given line to obtain a point B such that AB=5.6 units. From B, mark a distance of 1 unit and mark the new points as C. Find the midpoint of AC and mark the points as O. Draw a semicircle with centre O and radius OC. Draw a line perpendicular to AC passing through B and intersecting the semicircle at D. Then .
Now, draw an arc with centre B and radius BD, which intersects the number line in E.
Thus, E represent
If \(a+b\sqrt3=\dfrac{\sqrt31}{\sqrt3+1} \), find the values of \(a \) and \(b \).
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Express \(\dfrac{y^2}{\sqrt{x^2+y^2}+x} \) with rational denominator
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Find the values of \(x \) and \(y \) if \(\mathrm{(A)\ \dfrac{7+\sqrt5}{7\sqrt5}}\dfrac{7\sqrt5}{7+\sqrt5}=x+\dfrac{7}{11}\sqrt5y \).
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If \(x=\dfrac{\sqrt31}{\sqrt3+1} \) and \(y=\dfrac{3+2\sqrt2}{32\sqrt2} \) then find the value of \( x+y\).
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Express \(3.42\bar5 \) in the form \(\dfrac{p}{q} \), where \(p \) and \(q \) are integers, \(q\ne0 \)
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If \(\dfrac{\sqrt31}{\sqrt3+1}=a+b\sqrt3 \). find the values of a and b.
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If \(a=\dfrac{3\sqrt5}{3+\sqrt5} \) and \(b=\dfrac{3+\sqrt5}{3\sqrt5} \), find \( a^2b^2\).
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If \(\dfrac{\sqrt2\sqrt5}{\sqrt2+\sqrt5}=a+b\sqrt{10} \), find the values of a and b.
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If \(\dfrac{3+\sqrt7}{3\sqrt7}+\dfrac{3\sqrt7}{3+\sqrt7}=a+b\sqrt7 \), find the values of a and b.
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Find the value of \(a \), if \(\dfrac{3\sqrt5}{3+2\sqrt5}=a\sqrt5\dfrac{19}{11} \)
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If \(x=2+\sqrt5 \) find the value of \(\left(x^2\dfrac{1}{x^2}\right) \)
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If \(x=\dfrac{\sqrt5\sqrt2}{\sqrt5+\sqrt2} \) and \(y=\dfrac{\sqrt5+\sqrt2}{\sqrt5\sqrt2} \), find the value of \(x^2+xy+y^2 \)
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Simplify : \(\dfrac{\sqrt6}{\sqrt2+\sqrt3}+\dfrac{3\sqrt2}{\sqrt6+\sqrt3}\dfrac{4\sqrt3}{\sqrt6+\sqrt2} \).
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Express \(\dfrac{3}{\sqrt3\sqrt2+\sqrt5} \) with rational denominator
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If \(x=7+4\sqrt3 \) find the value of \(\left(x^3+\dfrac{1}{x^3}\right) \).
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If \(x=\dfrac{\sqrt3+\sqrt2}{\sqrt3\sqrt2} \) and \(y=\dfrac{\sqrt3\sqrt2}{\sqrt3+\sqrt2} \), then find the value of \(x^2+y^210xy \).
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Find the square root of 4.2 geometrically.
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Draw AC = 4.2
Take mid point as O
now draw a circle of OA = OC radius from centre O
Now take a point B from O at OB distance and make perpendicular D
Now BD is measure it
Show that \(\dfrac{1}{1+\sqrt2}+\dfrac{1}{\sqrt2+\sqrt3}=1+\sqrt3 \).
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If \(x=3+2\sqrt2 \), find the value of \(\left(\sqrt x\dfrac{1}{\sqrt x}\right) \)
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Find the square root of 4.5 geometrically.
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Step 1: Draw a number line. Mark O as the zero on the number line.
Step 2: Mark a point A as 4.5 on the number line.
Step 3: Mark a point C as 1 on the number line.
Step 4: Mark a point B as the midpoint of AC.
Step 5: With point B as the centre and radius as AB draw a semicircle.
Step 6: From O draw a perpendicular line to the number line that intersects the previous drawn semicircle at D. Here OD = square root of 4.5.
Step 7: With O as centre and radius as OD, draw an arc that intersects the number line at point E. Here E is the point square root of 4.5 on the number line.
If \(a = 2, b = 3\), then find the values of the following
(A) \((a^b+b^a)^{1} \) (B) \((a^a+b^b)^{1} \)
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Prove that \(\dfrac{1}{1+x^{ab}}+\dfrac{1}{1+x^{ba}} =1\).
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Express \(2.02\bar5 \) in the form of \(\dfrac{p}{q} \), where \(p \) and \(q \) are integers and \(q\ne0 \)
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Express \(0.6+0.\bar 7+0.4\bar7 \) in the form \(\dfrac{p}{q} \), where \(p,q\in Z,q\ne0 \).
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Let
Required sum
If \(a=5+2\sqrt6 \) and \(b=\dfrac{1}{a} \), then find the value of \(a^2+b^2 \).
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Find the value of : \(\dfrac{1}{1+\sqrt2}+\dfrac{1}{\sqrt2+\sqrt3}+\dfrac{1}{\sqrt3+\sqrt4}+…..\dfrac{1}{\sqrt8+\sqrt9} \)
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Find the values of \(a \) and \(b \) if : \(\dfrac{7+3\sqrt5}{3+\sqrt5}\dfrac{73\sqrt5}{3\sqrt5}=a+\sqrt{5b} \)
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If \(a=74\sqrt3 \), find the value of \(\left(\sqrt a+\dfrac{1}{\sqrt a}\right) \)
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Represent \(\sqrt{9.3} \) on the number line.
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Steps:
1) Draw a line segment AB of length 9.3 units.
2) Extend the line by 1 unit more such that BC=1 unit .
3) Find the midpoint of AC.
4) Draw a line BD perpendicular to AB and let it intersect the semicircle at point D.
5) Draw an arc DE such that BE = BD.
Therefore, BE = units