Verify the Associative Law Of Multiplication for the ratinoal number 3/-13,-5/7 and 9/23 ,Also verify the distrbutive law of multiplication over addition?
Answers
Answer:
GIVEN :
The rational numbers are -\frac{3}{13}−
13
3
, -\frac{5}{7}−
7
5
, \frac{9}{23}
23
9
TO VERIFY :
The associative law of multiplication for the given rational numbers and also verify the distributive property of addition over multiplication.
SOLUTION :
Given rational numbers are -\frac{3}{13}−
13
3
, -\frac{5}{7}−
7
5
, \frac{9}{23}
23
9
For any rational numbers a, b and c:
The Associative law over multiplication is given by
a\times (b\times c)=(a\times b)\times (a\times c)a×(b×c)=(a×b)×(a×c)
Let a=-\frac{3}{13}a=−
13
3
, b=-\frac{5}{7}b=−
7
5
, c=\frac{9}{23}c=
23
9
Now substituting the values in the formula,
Now verify the Associative law over multiplication
Now taking LHS
a\times (b\times c)a×(b×c)
-\frac{3}{13}\times (-\frac{5}{7}\times \frac{9}{23})−
13
3
×(−
7
5
×
23
9
)
=-\frac{3}{13}\times (\frac{-45}{161})=−
13
3
×(
161
−45
)
=\frac{135}{2093}=
2093
135
-\frac{3}{13}\times (-\frac{5}{7})\times \frac{9}{23})=\frac{12}{161}=0.0645−
13
3
×(−
7
5
)×
23
9
)=
161
12
=0.0645 =LHS
Now RHS (a\times b)\times c(a×b)×c
(-\frac{3}{13}\times (-\frac{5}{7}))\times \frac{9}{23}(−
13
3
×(−
7
5
))×
23
9
=\frac{15}{91}\times \frac{9}{23}=
91
15
×
23
9
=\frac{135}{2093}=
2093
135
=0.0645=RHS
∴ LHS = RHS
The Associative law over multiplication for the given rational numbers -\frac{3}{13}−
13
3
, -\frac{5}{7}−
7
5
, \frac{9}{23}
23
9
is verified.
∴ -\frac{3}{13}\times (-\frac{5}{7}\times \frac{9}{23})=((-\frac{3}{13})\times (-\frac{5}{7}))\times \frac{9}{23}−
13
3
×(−
7
5
×
23
9
)=((−
13
3
)×(−
7
5
))×
23
9
For any rational numbers a, b and c:
The Distributive property of addition over multiplication is given by
a\times (b+c)=a\times b+a\times ca×(b+c)=a×b+a×c
Let a=-\frac{3}{13}a=−
13
3
, b=-\frac{5}{7}b=−
7
5
, c=\frac{9}{23}c=
23
9
Now substituting the values in the formula,
Now verify the Distributive property of addition over multiplication
Now taking LHS
a\times (b+c)a×(b+c)
-\frac{3}{13}\times (-\frac{5}{7}+\frac{9}{23})−
13
3
×(−
7
5
+
23
9
)
=-\frac{3}{13}\times (\frac{-115+63}{161})=−
13
3
×(
161
−115+63
)
=-\frac{3}{13}\times (\frac{-52}{161})=−
13
3
×(
161
−52
)
=\frac{12}{161}=
161
12
-\frac{3}{13}\times (-\frac{5}{7}+\frac{9}{23})=\frac{12}{161}=0.0745−
13
3
×(−
7
5
+
23
9
)=
161
12
=0.0745 =LHS
Now RHS a\times b+a\times ca×b+a×c
-\frac{3}{13}\times (-\frac{5}{7})+(-\frac{3}{13})\times \frac{9}{23}−
13
3
×(−
7
5
)+(−
13
3
)×
23
9
=\frac{15}{91}-\frac{27}{299}=
91
15
−
299
27
=\frac{4485-2457}{27209}=
27209
4485−2457
=\frac{2028}{27209}=
27209
2028
=0.0745=RHS
∴ LHS = RHS
The Distributive property of addition over multiplication for the given rational numbers -\frac{3}{13}−
13
3
, -\frac{5}{7}−
7
5
, \frac{9}{23}
23
9
is verified.
∴ -\frac{3}{13}\times (-\frac{5}{7}+\frac{9}{23})=-\frac{3}{13}\times (-\frac{5}{7})+(-\frac{3}{13})\times \frac{9}{23}−
13
3
×(−
7
5
+
23
9
)=−
13
3
×(−
7
5
)+(−
13
3
)×
23
9
Step-by-step explanation:
this is the explaintaination
