To do: To give three examples of shapes with no line of symmetry.Solution: A line of symmetry is a line that divides an object into two equal pieces. When a figure is folded in half, the two halves coincide exactly along its line of symmetry. This line of symmetry is called the ... Read More

Given:$(i)$. $2^6$$(ii)$. $9^3$$(iii)$. $11^2$$(iv)$. $5^4$To do: To find the value of given terms.Solution:$(i)$. $2^6$$=2\times2\times2\times2\times2\times2$$=64$$(ii)$. $9^3$$=9\times9\times9$$=729$$(iii)$. $11^2$$=11\times11$$=121$$(iv)$. $5^4$$=5\times5\times5\times5$$=625$

Given: $(i)$. $6\times6\times6\times6$$(ii)$. $t\times t$$(iii)$. $b\times b\times b\times b$$(iv)$. $5\times5\times7\times7\times7$$(v)$. $2\times2\times a\times a$$(vi)$. $a\times a\times a\times c\times c\times c\times c\times d$To do:  To express the given terms in exponential form.Solution:$(i)$. $6\times6\times6\times6$$=6^4$$(ii)$. $t\times t$$=t^2$$(iii)$. $b\times b\times b\times b$$=b^4$$(iv)$. $5\times5\times7\times7\times7$$=5^{2\ }\times7^3$$(v)$. $2\times2\times a\times a$$=2^2\times a^2$$(vi)$. $a\times a\times a\times c\times c\times c\times c\times ... Read More

Given :The given numbers are $(i).\ 3125$ $(ii).\ 729$ $(iii).\ 343$ $(iv).\ 512$.To do :We have to express the given numbers in exponential form.Solution :$(i).\ 512$$512 = 8 \times 8\times 8 = 8^3$Therefore, 512 in exponential form is $8^3$.  $(ii).\ 343$$343 = 7 \times 7\times 7 = 7^3$Therefore, 343 in ... Read More

Given: $(i)$. $4^{3}$ or $3^4$$(ii)$. $5^3$ or $3^5$$(iii)$. $2^8$ or $8^2$$(iv)$. $100^2$ or $2^{100}$$(v)$. $2^{10}$ or $10^2$To do: To identify the greater number, wherever possible, in each of the cases.Solution:  $(i)$. $4^3$ or $3^4$$4^3=4\times4\times4$$=64$Now, $3^4=3\times3\times3\times3$$=81$Since, $64$$100$So, $2^{10}$>$10^2$

Given: $(i)$. $2\times10^3$$(ii)$. $7^2\times2^2$$(iii)$. $2^3\times5$$(iv)$. $3\times4^4$$(v)$. $0\times10^2$$(vi)$. $5^2\times3^3$$(viii)$. $2^4\times3^2$$(viii)$. $3^2\times10^4$To do: To simplify the given numbers.Solution:$(i)$. $2\times10\times10\times10=2000$$(ii)$. $7\times7\times2\times2=196$$(iii)$. $2\times2\times2\times5=40$$(iv)$. $3\times4\times4\times4\times4=768$$(v)$. $0\times10\times10=0$$(vi)$. $5\times5\times3\times3\times3=675$$(viii)$. $2\times2\times2\times2\times3\times3=144$$(viii)$. $3\times3\times10\times10\times10\times10=90000$Read More

Given:$(i)$. $(-4)^3$$(ii)$. $(-3)\times(-2)^3$$(iii)$. $(-3)^2\times(-5)^2$$(iv)$. $(-2)^3\times(-10)^3$To do: To simplify the given terms.Solution:$(i)$. $(-4)^3$$=-4\times-4\times-4$$=-64$$(ii)$. $(-3)\times(-2)^3$$=(-3)\times(-2)\times(-2)\times(-2)$$=24$$(iii)$. $(-3)^2\times(-5)^2$$=(-3)\times(-3)\times(-5)\times(-5)$$=225$$(iv)$. $(-2)^3\times(-10)^3$$=(-2)\times(-2)\times(-2)\times(-10)\times(-10)\times(-10)$$=8000$Read More

Given:Given numbers are$(i)$. $2.7\times10^{12};\ 1.5\times10^8$$(ii)$. $4\times10^{14};\ 3\times10^{17}$To do:We have to compare the given numbers.Solution:$(i)$. $2.7 \times 10^{12} = 2.7 \times 10^8 \times 10^4$$=(2.7\times10000)\times10^8$$=27000\times10^8$As $27000>1.5$$27000\times10^8 > 1.5\times10^8$Therefore,$2.7 \times 10^{12} > 1.5\times10^8$.$(ii)$. $3 \times 10^{17} = 3 \times 10^3 \times 10^{14}$$=(3\times1000)\times10^{14}$$=3000\times10^{14}$As $3000>4$$3000\times10^{14} > 4\times10^{14}$Therefore,$3 \times 10^{17} > 4\times10^{14}$.

To do:We have to classify the given numbers as rational or irrational.Solution: (i) We know that, $\sqrt{5}=2.236067..........$The decimal expansion of \( \sqrt{5} \) is non-terminating and non-recurring.Therefore, \( 2-\sqrt{5} \) is an irrational number.(ii) $(3+\sqrt{23})-\sqrt{23}=3+\sqrt{23}-\sqrt{23}$$=3$$=\frac{3}{1}$The number $\frac{3}{1}$ is in $\frac{p}{q}$ form.Hence, \( (3+\sqrt{23})-\sqrt{23} \) is a rational number.(iii) \( \frac{2 \sqrt{7}}{7 \sqrt{7}}=\frac{2}{7} ... Read More

To do: We have to simplify the given expressions.Solution:We know that, $(a+b)(a-b)=a^2-b^2$$(a+b)^2=a^2+2ab+b^2$$(a-b)^2=a^2-2ab+b^2$$(a+b)(c+d)=a(c+d)+b(c+d)$Therefore, (i) $(3+\sqrt{3})(2+\sqrt{2})=3(2+\sqrt2)+\sqrt3(2+\sqrt2)$$=3(2)+3\times \sqrt2+\sqrt3 \times2+\sqrt3 \times \sqrt2$$=6+3\sqrt2+2\sqrt3+\sqrt{3\times2}$$=6+3\sqrt2+2\sqrt3+\sqrt6$(ii) $(3+\sqrt{3})(3-\sqrt{3})=(3)^2-(\sqrt3)^2$$=9-3$$=6$(iii) $(\sqrt{5}+\sqrt{3})^{2}=(\sqrt{5})^{2}+(\sqrt{3})^{2}+2 \times \sqrt{5} \times \sqrt{3}$$=5+3+2 \sqrt{5\times3}$$=8+2 \sqrt{15}$ (iv) $(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})=(\sqrt5)^2-(\sqrt2)^2$$=5-2$$=3$Read More